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Theorem List for Metamath Proof Explorer - 34201-34300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembnj1307 34201* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( w  e.  B  ->  A. x  w  e.  B )   =>    |-  ( w  e.  C  ->  A. x  w  e.  C )
 
Theorembnj1311 34202* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  D  =  ( dom  g  i^i 
 dom  h )   =>    |-  ( ( R 
 FrSe  A  /\  g  e.  C  /\  h  e.  C )  ->  (
 g  |`  D )  =  ( h  |`  D ) )
 
Theorembnj1318 34203 Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( X  =  Y  ->  trCl
 ( X ,  A ,  R )  =  trCl ( Y ,  A ,  R ) )
 
Theorembnj1326 34204* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  D  =  ( dom  g  i^i 
 dom  h )   =>    |-  ( ( R 
 FrSe  A  /\  g  e.  C  /\  h  e.  C )  ->  (
 g  |`  D )  =  ( h  |`  D ) )
 
Theorembnj1321 34205* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   =>    |-  ( ( R  FrSe  A 
 /\  E. f ta )  ->  E! f ta )
 
Theorembnj1364 34206 Property of  FrSe. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( R  FrSe  A  ->  R  Se  A )
 
Theorembnj1371 34207* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  ( ta'  <->  ( f  e.  C  /\  dom  f  =  ( {
 y }  u.  trCl ( y ,  A ,  R ) ) ) )   =>    |-  ( f  e.  H  ->  Fun  f )
 
Theorembnj1373 34208* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  ( ta'  <->  [. y  /  x ].
 ta )   =>    |-  ( ta'  <->  ( f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
 
Theorembnj1374 34209* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   =>    |-  ( f  e.  H  ->  f  e.  C )
 
Theorembnj1384 34210* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   =>    |-  ( R  FrSe  A  ->  Fun  P )
 
Theorembnj1388 34211* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   =>    |-  ( ch  ->  A. y  e.  pred  ( x ,  A ,  R ) E. f ta' )
 
Theorembnj1398 34212* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  ( th 
 <->  ( ch  /\  z  e.  U_ y  e.  pred  ( x ,  A ,  R ) ( {
 y }  u.  trCl ( y ,  A ,  R ) ) ) )   &    |-  ( et  <->  ( th  /\  y  e.  pred ( x ,  A ,  R )  /\  z  e.  ( { y }  u.  trCl
 ( y ,  A ,  R ) ) ) )   =>    |-  ( ch  ->  U_ y  e.  pred  ( x ,  A ,  R )
 ( { y }  u.  trCl ( y ,  A ,  R ) )  =  dom  P )
 
Theorembnj1413 34213* Property of  trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  (  pred ( X ,  A ,  R )  u.  U_ y  e.  pred  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) )   =>    |-  ( ( R  FrSe  A 
 /\  X  e.  A )  ->  B  e.  _V )
 
Theorembnj1408 34214* Technical lemma for bnj1414 34215. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  (  pred ( X ,  A ,  R )  u.  U_ y  e.  pred  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) )   &    |-  C  =  ( 
 pred ( X ,  A ,  R )  u.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A ) )   &    |-  ( ta  <->  ( B  e.  _V 
 /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) )   =>    |-  ( ( R 
 FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  =  B )
 
Theorembnj1414 34215* Property of  trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  (  pred ( X ,  A ,  R )  u.  U_ y  e.  pred  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) )   =>    |-  ( ( R  FrSe  A 
 /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  =  B )
 
Theorembnj1415 34216* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   =>    |-  ( ch  ->  dom  P  =  trCl ( x ,  A ,  R ) )
 
Theorembnj1416 34217 Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  ( ch  ->  dom 
 P  =  trCl ( x ,  A ,  R ) )   =>    |-  ( ch  ->  dom 
 Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
 
Theorembnj1418 34218 Property of  pred. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 y  e.  pred ( x ,  A ,  R )  ->  y R x )
 
Theorembnj1417 34219* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
 |-  ( ph 
 <->  R  FrSe  A )   &    |-  ( ps 
 <->  -.  x  e.  trCl ( x ,  A ,  R ) )   &    |-  ( ch 
 <-> 
 A. y  e.  A  ( y R x 
 ->  [. y  /  x ].
 ps ) )   &    |-  ( th 
 <->  ( ph  /\  x  e.  A  /\  ch )
 )   &    |-  B  =  (  pred ( x ,  A ,  R )  u.  U_ y  e.  pred  ( x ,  A ,  R )  trCl ( y ,  A ,  R ) )   =>    |-  ( ph  ->  A. x  e.  A  -.  x  e.  trCl ( x ,  A ,  R ) )
 
Theorembnj1421 34220* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  ( ch  ->  Fun 
 P )   &    |-  ( ch  ->  dom 
 Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )   &    |-  ( ch  ->  dom  P  =  trCl ( x ,  A ,  R ) )   =>    |-  ( ch  ->  Fun 
 Q )
 
Theorembnj1444 34221* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  P  Fn  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( th  <->  ( ch  /\  z  e.  E )
 )   &    |-  ( et  <->  ( th  /\  z  e.  { x } ) )   &    |-  ( ze 
 <->  ( th  /\  z  e.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( rh  <->  ( ze  /\  f  e.  H  /\  z  e.  dom  f ) )   =>    |-  ( rh  ->  A. y rh )
 
Theorembnj1445 34222* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  P  Fn  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( th  <->  ( ch  /\  z  e.  E )
 )   &    |-  ( et  <->  ( th  /\  z  e.  { x } ) )   &    |-  ( ze 
 <->  ( th  /\  z  e.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( rh  <->  ( ze  /\  f  e.  H  /\  z  e.  dom  f ) )   &    |-  ( si  <->  ( rh  /\  y  e.  pred ( x ,  A ,  R )  /\  f  e.  C  /\  dom  f  =  ( { y }  u.  trCl
 ( y ,  A ,  R ) ) ) )   &    |-  ( ph  <->  ( si  /\  d  e.  B  /\  f  Fn  d  /\  A. x  e.  d  (
 f `  x )  =  ( G `  Y ) ) )   &    |-  X  =  <. z ,  (
 f  |`  pred ( z ,  A ,  R ) ) >.   =>    |-  ( si  ->  A. d si )
 
Theorembnj1446 34223* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   =>    |-  ( ( Q `
  z )  =  ( G `  W )  ->  A. d ( Q `
  z )  =  ( G `  W ) )
 
Theorembnj1447 34224* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   =>    |-  ( ( Q `
  z )  =  ( G `  W )  ->  A. y ( Q `
  z )  =  ( G `  W ) )
 
Theorembnj1448 34225* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   =>    |-  ( ( Q `
  z )  =  ( G `  W )  ->  A. f ( Q `
  z )  =  ( G `  W ) )
 
Theorembnj1449 34226* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  P  Fn  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( th  <->  ( ch  /\  z  e.  E )
 )   &    |-  ( et  <->  ( th  /\  z  e.  { x } ) )   &    |-  ( ze 
 <->  ( th  /\  z  e.  trCl ( x ,  A ,  R )
 ) )   =>    |-  ( ze  ->  A. f ze )
 
Theorembnj1442 34227* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  P  Fn  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( th  <->  ( ch  /\  z  e.  E )
 )   &    |-  ( et  <->  ( th  /\  z  e.  { x } ) )   =>    |-  ( et  ->  ( Q `  z )  =  ( G `  W ) )
 
Theorembnj1450 34228* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  P  Fn  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( th  <->  ( ch  /\  z  e.  E )
 )   &    |-  ( et  <->  ( th  /\  z  e.  { x } ) )   &    |-  ( ze 
 <->  ( th  /\  z  e.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( rh  <->  ( ze  /\  f  e.  H  /\  z  e.  dom  f ) )   &    |-  ( si  <->  ( rh  /\  y  e.  pred ( x ,  A ,  R )  /\  f  e.  C  /\  dom  f  =  ( { y }  u.  trCl
 ( y ,  A ,  R ) ) ) )   &    |-  ( ph  <->  ( si  /\  d  e.  B  /\  f  Fn  d  /\  A. x  e.  d  (
 f `  x )  =  ( G `  Y ) ) )   &    |-  X  =  <. z ,  (
 f  |`  pred ( z ,  A ,  R ) ) >.   =>    |-  ( ze  ->  ( Q `  z )  =  ( G `  W ) )
 
Theorembnj1423 34229* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  P  Fn  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R )
 ) )   =>    |-  ( ch  ->  A. z  e.  E  ( Q `  z )  =  ( G `  W ) )
 
Theorembnj1452 34230* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   =>    |-  ( ch  ->  E  e.  B )
 
Theorembnj1466 34231* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   =>    |-  ( w  e.  Q  ->  A. f  w  e.  Q )
 
Theorembnj1467 34232* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   =>    |-  ( w  e.  Q  ->  A. d  w  e.  Q )
 
Theorembnj1463 34233* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  Q  e.  _V )   &    |-  ( ch  ->  A. z  e.  E  ( Q `  z )  =  ( G `  W ) )   &    |-  ( ch  ->  Q  Fn  E )   &    |-  ( ch  ->  E  e.  B )   =>    |-  ( ch  ->  Q  e.  C )
 
Theorembnj1489 34234* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   =>    |-  ( ch  ->  Q  e.  _V )
 
Theorembnj1491 34235* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  ( ch  ->  ( Q  e.  C  /\  dom 
 Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )   =>    |-  ( ( ch  /\  Q  e.  _V )  ->  E. f ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
 
Theorembnj1312 34236* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   =>    |-  ( R  FrSe  A  ->  A. x  e.  A  E. f  e.  C  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
 
Theorembnj1493 34237* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   =>    |-  ( R  FrSe  A  ->  A. x  e.  A  E. f  e.  C  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R )
 ) )
 
Theorembnj1497 34238* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   =>    |-  A. g  e.  C  Fun  g
 
Theorembnj1498 34239* Technical lemma for bnj60 34240. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   =>    |-  ( R  FrSe  A  ->  dom 
 F  =  A )
 
21.28.5  Well-founded recursion, part 1 of 3
 
Theorembnj60 34240* Well-founded recursion, part 1 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   =>    |-  ( R  FrSe  A  ->  F  Fn  A )
 
Theorembnj1514 34241* Technical lemma for bnj1500 34246. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   =>    |-  (
 f  e.  C  ->  A. x  e.  dom  f
 ( f `  x )  =  ( G `  Y ) )
 
Theorembnj1518 34242* Technical lemma for bnj1500 34246. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   &    |-  ( ph  <->  ( R  FrSe  A 
 /\  x  e.  A ) )   &    |-  ( ps  <->  ( ph  /\  f  e.  C  /\  x  e. 
 dom  f ) )   =>    |-  ( ps  ->  A. d ps )
 
Theorembnj1519 34243* Technical lemma for bnj1500 34246. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   =>    |-  ( ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. )  ->  A. d
 ( F `  x )  =  ( G ` 
 <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. ) )
 
Theorembnj1520 34244* Technical lemma for bnj1500 34246. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   =>    |-  ( ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. )  ->  A. f
 ( F `  x )  =  ( G ` 
 <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. ) )
 
Theorembnj1501 34245* Technical lemma for bnj1500 34246. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   &    |-  ( ph  <->  ( R  FrSe  A 
 /\  x  e.  A ) )   &    |-  ( ps  <->  ( ph  /\  f  e.  C  /\  x  e. 
 dom  f ) )   &    |-  ( ch  <->  ( ps  /\  d  e.  B  /\  dom  f  =  d ) )   =>    |-  ( R  FrSe  A  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. ) )
 
21.28.6  Well-founded recursion, part 2 of 3
 
Theorembnj1500 34246* Well-founded recursion, part 2 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   =>    |-  ( R  FrSe  A  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. ) )
 
Theorembnj1525 34247* Technical lemma for bnj1522 34250. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   &    |-  ( ph  <->  ( R  FrSe  A 
 /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R )
 ) >. ) ) )   &    |-  ( ps  <->  ( ph  /\  F  =/=  H ) )   =>    |-  ( ps  ->  A. x ps )
 
Theorembnj1529 34248* Technical lemma for bnj1522 34250. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ch  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. ) )   &    |-  ( w  e.  F  ->  A. x  w  e.  F )   =>    |-  ( ch  ->  A. y  e.  A  ( F `  y )  =  ( G `  <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >. ) )
 
Theorembnj1523 34249* Technical lemma for bnj1522 34250. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   &    |-  ( ph  <->  ( R  FrSe  A 
 /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R )
 ) >. ) ) )   &    |-  ( ps  <->  ( ph  /\  F  =/=  H ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  A  /\  ( F `
  x )  =/=  ( H `  x ) ) )   &    |-  D  =  { x  e.  A  |  ( F `  x )  =/=  ( H `  x ) }   &    |-  ( th 
 <->  ( ch  /\  y  e.  D  /\  A. z  e.  D  -.  z R y ) )   =>    |-  ( ( R 
 FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R )
 ) >. ) )  ->  F  =  H )
 
21.28.7  Well-founded recursion, part 3 of 3
 
Theorembnj1522 34250* Well-founded recursion, part 3 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   =>    |-  ( ( R  FrSe  A 
 /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R )
 ) >. ) )  ->  F  =  H )
 
21.29  Mathbox for BJ

In this mathbox, we try to respect the ordering of the sections of the main part. There are strengthenings of theorems of the main part, as well as work on reducing axiom dependencies.

 
21.29.1  Propositional calculus

Miscellaneous utility theorems of propositional calculus.

 
21.29.1.1  Derived rules of inference

In this section, we prove a few rules of inference derived from modus ponens, and which do not depend on any axioms.

 
Theorembj-mp2c 34251 A double modus ponens inference. (Contributed by BJ, 24-Sep-2019.)
 |-  ph   &    |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ps  ->  ch )
 )   =>    |- 
 ch
 
Theorembj-mp2d 34252 A double modus ponens inference. (Contributed by BJ, 24-Sep-2019.)
 |-  ph   &    |-  ( ph  ->  ps )   &    |-  ( ps  ->  (
 ph  ->  ch ) )   =>    |-  ch
 
21.29.1.2  A syntactic theorem

In this section, we prove a syntactic theorem (bj-0 34253) asserting that some formula is indeed a formula. Then, we use this syntactic theorem to shorten the proof of a "usual" theorem (bj-1 34254) and explain in the comment of that theorem why this phenomenon is unusual.

 
Theorembj-0 34253 A syntactic theorem. See the comment of bj-1 34254. The only other syntactic theorems in the main part of set.mm are wel 1820 and weq 1734. (Contributed by BJ, 24-Sep-2019.) (New usage is discouraged.)
 wff  ( ( ph  ->  ps )  ->  ch )
 
Theorembj-1 34254 In this proof, the use of the syntactic theorem bj-0 34253 allows to reduce the length of the proof by one (non-essential) step. Since bj-0 34253 is used in a non-essential step, this use does not appear in the html page (but the present theorem appears on the html page of bj-0 34253 as a theorem using it.

Suppose that the syntactic theorem thm0 proves that PHI is a formula, and that thm0 is used to shorten the proof of thm1. Assume that PHI does have proper non-atomic subformulas of type wff (which is not the case of the formula proved by weq 1734 or wel 1820). Then, the proof of thm1 does not construct all the proper non-atomic subformulas of PHI (if it did, then using thm0 would not shorten it). Therefore, thm1 is a special instance of a more general theorem with essentially the same proof. This explains why syntactic theorems are not useful in set.mm. In the present case, bj-1 34254 is a special instance of id 22. (Contributed by BJ, 24-Sep-2019.) (New usage is discouraged.)

 |-  (
 ( ( ph  ->  ps )  ->  ch )  ->  ( ( ph  ->  ps )  ->  ch )
 )
 
21.29.1.3  Syllogisms

The three theorems sylbbr 34255, sylbb1 34256 and sylbb2 34257 could be placed, together with sylbb 197, after the theorems sylibr 212 and sylbir 213. Together with them and the theorems sylib 196 and sylbi 195, they are the 8 possible inferences inferring an implication from an implication and a biconditional (4) or from two biconditionals (4). The 4 inferences proving a biconditional from two biconditionals are bitri 249, bitr2i 250, bitr3i 251 and bitr4i 252, while the inference proving an implication from two implications is syl 16 (for consistency, it could be labelled "imtri" but "syl" is classical). There are also closed forms and deduction versions of these, like, among many others, syld 44, syl5 32, syl6 33, mpbid 210, bitrd 253, syl5bb 257, syl6bb 261 and variants.

Additionally, the three newly added theorems help shorten 17 existing proofs by a total of around 250 bytes.

 
Theoremsylbbr 34255 A mixed syllogism inference from two biconditionals. (Minimizes bitri 249.) (Contributed by BJ, 21-Apr-2019.)
 |-  ( ph 
 <->  ps )   &    |-  ( ps  <->  ch )   =>    |-  ( ch  ->  ph )
 
Theoremsylbb1 34256 A mixed syllogism inference from two biconditionals. (Contributed by BJ, 21-Apr-2019.)
 |-  ( ph 
 <->  ps )   &    |-  ( ph  <->  ch )   =>    |-  ( ps  ->  ch )
 
Theoremsylbb2 34257 A mixed syllogism inference from two biconditionals. (Contributed by BJ, 21-Apr-2019.)
 |-  ( ph 
 <->  ps )   &    |-  ( ch  <->  ps )   =>    |-  ( ph  ->  ch )
 
21.29.1.4  Implication and negation
 
Theorempm4.81ALT 34258 Alternate proof of pm4.81 366. (Contributed by BJ, 30-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( -.  ph  ->  ph )  <->  ph )
 
Theorembj-jarri 34259 Inference associated with jarr 98. (Contributed by BJ, 29-Mar-2020.)
 |-  (
 ( ph  ->  ps )  ->  ch )   =>    |-  ( ps  ->  ch )
 
Theorembj-con4iALT 34260 Alternate proof of con4i 130. Probably the original proof. (Contributed by BJ, 29-Mar-2020.)
 |-  ( -.  ph  ->  -.  ps )   =>    |-  ( ps  ->  ph )
 
Theorembj-con2com 34261 A commuted form of the contrapositive, true in minimal calculus. (Contributed by BJ, 19-Mar-2020.)
 |-  ( ph  ->  ( ( ps 
 ->  -.  ph )  ->  -.  ps ) )
 
Theorembj-con2comi 34262 Inference associated with bj-con2com 34261. Its associated inference is mt2 179. TODO: when in the main part, add to mt2 179 that it is the inference associated with bj-con2comi 34262. (Contributed by BJ, 19-Mar-2020.)
 |-  ph   =>    |-  ( ( ps  ->  -.  ph )  ->  -.  ps )
 
Theorembj-pm2.18i 34263 Inference associated with pm2.18 110. (Contributed by BJ, 30-Mar-2020.)
 |-  ( -.  ph  ->  ph )   =>    |-  ph
 
Theorembj-pm2.01i 34264 Inference associated with pm2.01 168. (Contributed by BJ, 30-Mar-2020.)
 |-  ( ph  ->  -.  ph )   =>    |-  -.  ph
 
Theorembj-nimn 34265 If a formula is true, then it does not imply its negation. (Contributed by BJ, 19-Mar-2020.) A shorter proof is possible using id 22 and jc 147, however, the present proof uses theorems that are more basic than jc 147. (Proof modification is discouraged.)
 |-  ( ph  ->  -.  ( ph  ->  -.  ph ) )
 
Theorembj-nimni 34266 Inference associated with bj-nimn 34265. (Contributed by BJ, 19-Mar-2020.)
 |-  ph   =>    |- 
 -.  ( ph  ->  -.  ph )
 
Theorembj-peircei 34267 Inference associated with peirce 181. (Contributed by BJ, 30-Mar-2020.)
 |-  (
 ( ph  ->  ps )  -> 
 ph )   =>    |-  ph
 
Theorembj-looinvi 34268 Inference associated with looinv 182. (Contributed by BJ, 30-Mar-2020.)
 |-  (
 ( ph  ->  ps )  ->  ps )   =>    |-  ( ( ps  ->  ph )  ->  ph )
 
Theorembj-looinvii 34269 Inference associated with bj-looinvi 34268. (Contributed by BJ, 30-Mar-2020.)
 |-  (
 ( ph  ->  ps )  ->  ps )   &    |-  ( ps  ->  ph )   =>    |-  ph
 
21.29.1.5  Disjunction

A few lemmas about disjunction. The fundamental theorems in this family are the dual statements pm4.71 630 and pm4.72 876. See also biort 896 and biorf 405.

 
Theorembj-jaoi1 34270 Shortens 11 proofs by a total of around 60 bytes. (Contributed by BJ, 30-Sep-2019.)
 |-  ( ph  ->  ps )   =>    |-  ( ( ph  \/  ps )  ->  ps )
 
Theorembj-jaoi2 34271 Shortens 9 proofs by a total of around 50 bytes. (Contributed by BJ, 30-Sep-2019.)
 |-  ( ph  ->  ps )   =>    |-  ( ( ps  \/  ph )  ->  ps )
 
Theorembj-animorl 34272 Conjunction implies disjunction with one common formula (1/4). (Contributed by BJ, 4-Oct-2019.)
 |-  (
 ( ph  /\  ps )  ->  ( ph  \/  ch ) )
 
Theorembj-animorr 34273 Conjunction implies disjunction with one common formula (2/4). (Contributed by BJ, 4-Oct-2019.)
 |-  (
 ( ph  /\  ps )  ->  ( ch  \/  ps ) )
 
Theorembj-animorlr 34274 Conjunction implies disjunction with one common formula (3/4). (Contributed by BJ, 4-Oct-2019.)
 |-  (
 ( ph  /\  ps )  ->  ( ch  \/  ph ) )
 
Theorembj-animorrl 34275 Conjunction implies disjunction with one common formula (4/4). (Contributed by BJ, 4-Oct-2019.)
 |-  (
 ( ph  /\  ps )  ->  ( ps  \/  ch ) )
 
21.29.1.6  Logical equivalence

A few other characterizations of the bicondional. The inter-definability of logical connectives offers many ways to express a given statement. Some useful theorems in this regard are df-or 370, df-an 371, pm4.64 372, imor 412, pm4.62 419 through pm4.67 428, and, for the De Morgan laws, ianor 488 through pm4.57 497.

 
Theorembj-dfbi4 34276 Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
 |-  (
 ( ph  <->  ps )  <->  ( ( ph  /\ 
 ps )  \/  -.  ( ph  \/  ps )
 ) )
 
Theorembj-dfbi5 34277 Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
 |-  (
 ( ph  <->  ps )  <->  ( ( ph  \/  ps )  ->  ( ph  /\  ps ) ) )
 
Theorembj-dfbi6 34278 Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
 |-  (
 ( ph  <->  ps )  <->  ( ( ph  \/  ps )  <->  ( ph  /\  ps ) ) )
 
Theorembj-bijust0 34279 The general statement that bijust 183 proves (with a shorter proof). (Contributed by NM, 11-May-1999.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (Revised by BJ, 19-Mar-2020.)
 |-  -.  ( ( ph  ->  ph )  ->  -.  ( ph  ->  ph ) )
 
21.29.1.7  The conditional operator for propositions
 
Theoremanifp 34280 The conditional operator is implied by the conjunction of its possible outputs. Dual statement of ifpor 34281. (Contributed by BJ, 30-Sep-2019.)
 |-  (
 ( ps  /\  ch )  -> if- ( ph ,  ps ,  ch ) )
 
Theoremifpor 34281 The conditional operator implies the disjunction of its possible outputs. Dual statement of anifp 34280. (Contributed by BJ, 1-Oct-2019.)
 |-  (if- ( ph ,  ps ,  ch )  ->  ( ps 
 \/  ch ) )
 
Theorembj-elimhyp 34282 Version of elimh 956 expressed using the conditional operator. (Contributed by BJ, 30-Sep-2019.)
 |-  (
 (if- ( ch ,  ph ,  ps )  <->  ph )  ->  ( ch 
 <->  ta ) )   &    |-  (
 (if- ( ch ,  ph ,  ps )  <->  ps )  ->  ( th 
 <->  ta ) )   &    |-  th   =>    |-  ta
 
Theorembj-dedthm 34283 Version of dedt 957 expressed using the conditional operator. (Contributed by BJ, 30-Sep-2019.)
 |-  (
 (if- ( ch ,  ph ,  ps )  <->  ph )  ->  ( th 
 <->  ta ) )   &    |-  ta   =>    |-  ( ch  ->  th )
 
Theorembj-con3thALT 34284 Version of con3th 958 using the conditional operator in its proof. (Contributed by BJ, 30-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ps )  ->  ( -.  ps  ->  -.  ph ) )
 
Theorembj-consensus 34285 Version of consensus 959 expressed using the conditional operator. (Remark: it may be better to express it as consensus 959, using only binary connectives, and hinting at the fact that it is a Boolean algebra identity, like the absorption identities.) (Contributed by BJ, 30-Sep-2019.)
 |-  (
 (if- ( ph ,  ps ,  ch )  \/  ( ps  /\  ch ) )  <-> if- ( ph ,  ps ,  ch ) )
 
Theorembj-dfifc2 34286* This should be the alternate definition of "ifc" if "if-" enters the main part. (Contributed by BJ, 20-Sep-2019.)
 |-  if ( ph ,  A ,  B )  =  { x  |  ( ( ph  /\  x  e.  A )  \/  ( -.  ph  /\  x  e.  B ) ) }
 
Theorembj-df-ifc 34287* The definition of "ifc" if "if-" enters the main part. This is in line with the definition of a class as the extension of a predicate in df-clab 2443. (Contributed by BJ, 20-Sep-2019.)
 |-  if ( ph ,  A ,  B )  =  { x  | if- ( ph ,  x  e.  A ,  x  e.  B ) }
 
Theorembj-ififc 34288* A theorem linking if- and  if. (Contributed by BJ, 24-Sep-2019.)
 |-  ( x  e.  if ( ph ,  A ,  B ) 
 <-> if- ( ph ,  x  e.  A ,  x  e.  B ) )
 
21.29.1.8  Miscellaneous

Miscellaneous utility theorems of propositional calculus.

 
Theorembj-imim2ALT 34289 More direct proof of imim2 53. Note that imim2i 14 and imim2d 52 can be proved as usual from this closed form (i.e. using ax-mp 5 and syl 16 respectively). (Contributed by BJ, 19-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ps )  ->  ( ( ch  ->  ph )  ->  ( ch  ->  ps ) ) )
 
Theoremsylancl2 34290 Shortens 5 proofs. (Contributed by BJ, 25-Apr-2019.)
 |-  ( ph  ->  ps )   &    |-  ch   &    |-  ( ( ps 
 /\  ch )  <->  th )   =>    |-  ( ph  ->  th )
 
Theoremsylancl3 34291 Shortens 11 proofs by a total of around 150 bytes. (Contributed by BJ, 25-Apr-2019.)
 |-  ( ph  ->  ps )   &    |-  ch   &    |-  ( th  <->  ( ps  /\  ch ) )   =>    |-  ( ph  ->  th )
 
Theorembj-imbi12 34292 Imported form (uncurried form) of imbi12 322. (Contributed by BJ, 6-May-2019.)
 |-  (
 ( ( ph  <->  ps )  /\  ( ch 
 <-> 
 th ) )  ->  ( ( ph  ->  ch )  <->  ( ps  ->  th ) ) )
 
Theorembj-trut 34293 A proposition is equivalent to it being implied by T.. Closed form of trud 1404 (which it can shorten); dual of dfnot 1414. It is to tbtru 1405 what a1bi 337 is to tbt 344, and this appears in their respective proofs. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
 |-  ( ph 
 <->  ( T.  ->  ph )
 )
 
Theorembj-biorfi 34294 This should be labeled "biorfi" while the current biorfi 407 should be labelled "biorfri". The dual of biorf 405 is not biantr 931 but iba 503 (and ibar 504). So there should also be a "biorfr". (Note that these four statements can actually be strengthened to biconditionals.) (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
 |-  -.  ph   =>    |-  ( ps  <->  ( ph  \/  ps ) )
 
Theorembj-falor 34295 Dual of truan 1412 (which has biconditional reversed). (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
 |-  ( ph 
 <->  ( F.  \/  ph ) )
 
Theorembj-falor2 34296 Dual of truan 1412. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
 |-  (
 ( F.  \/  ph ) 
 <-> 
 ph )
 
Theorembj-bibibi 34297 A property of the biconditional. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
 |-  ( ph 
 <->  ( ps  <->  ( ph  <->  ps ) ) )
 
Theorembj-imn3ani 34298 Duplication of bnj1224 33982. Three-fold version of imnani 423. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by BJ, 22-Oct-2019.) (Proof modification is discouraged.)
 |-  -.  ( ph  /\  ps  /\  ch )   =>    |-  ( ( ph  /\  ps )  ->  -.  ch )
 
Theorembj-andnotim 34299 Two ways of expressing a certain ternary connective. Note the respective positions of the three wff's. (Contributed by BJ, 6-Oct-2018.)
 |-  (
 ( ( ph  /\  -.  ps )  ->  ch )  <->  ( ( ph  ->  ps )  \/  ch ) )
 
Theorembj-bi3ant 34300 This used to be in the main part. (Contributed by Wolf Lammen, 14-May-2013.) (Revised by BJ, 14-Jun-2019.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ( ( th  ->  ta )  ->  ph )  ->  ( ( ( ta 
 ->  th )  ->  ps )  ->  ( ( th  <->  ta )  ->  ch )
 ) )
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