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Theorem List for Metamath Proof Explorer - 29101-29200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembrpprod3a 29101* Condition for parallel product when the last argument is not an ordered pair. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  X  e.  _V   &    |-  Y  e.  _V   &    |-  Z  e.  _V   =>    |-  ( <. X ,  Y >.pprod ( R ,  S ) Z  <->  E. z E. w ( Z  =  <. z ,  w >.  /\  X R z  /\  Y S w ) )
 
Theorembrpprod3b 29102* Condition for parallel product when the first argument is not an ordered pair. (Contributed by Scott Fenton, 3-May-2014.)
 |-  X  e.  _V   &    |-  Y  e.  _V   &    |-  Z  e.  _V   =>    |-  ( Xpprod ( R ,  S ) <. Y ,  Z >.  <->  E. z E. w ( X  =  <. z ,  w >.  /\  z R Y  /\  w S Z ) )
 
Theoremrelsset 29103 The subset class is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  Rel  SSet
 
Theorembrsset 29104 For sets, the  SSet binary relationship is equivalent to the subset relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  B  e.  _V   =>    |-  ( A SSet B  <->  A 
 C_  B )
 
Theoremidsset 29105  _I is equal to  SSet and its converse. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  _I  =  ( SSet  i^i  `' SSet )
 
Theoremeltrans 29106 Membership in the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
 |-  A  e.  _V   =>    |-  ( A  e.  Trans  <->  Tr  A )
 
Theoremdfon3 29107 A quantifier-free definition of  On. (Contributed by Scott Fenton, 5-Apr-2012.)
 |-  On  =  ( _V  \  ran  ( ( SSet  i^i  ( Trans  X.  _V ) ) 
 \  (  _I  u.  _E  ) ) )
 
Theoremdfon4 29108 Another quantifier-free definition of  On. (Contributed by Scott Fenton, 4-May-2014.)
 |-  On  =  ( _V  \  (
 ( SSet  \  (  _I 
 u.  _E  ) ) "
 Trans ) )
 
Theorembrtxpsd 29109* Expansion of a common form used in quantifier-free definitions. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( -.  A ran  ( ( _V  (x)  _E  )(++) ( R  (x)  _V )
 ) B  <->  A. x ( x  e.  B  <->  x R A ) )
 
Theorembrtxpsd2 29110* Another common abbreviation for quantifier-free definitions. (Contributed by Scott Fenton, 21-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  R  =  ( C  \  ran  ( ( _V  (x)  _E  )(++) ( S  (x)  _V ) ) )   &    |-  A C B   =>    |-  ( A R B  <->  A. x ( x  e.  B  <->  x S A ) )
 
Theorembrtxpsd3 29111* A third common abbreviation for quantifier-free definitions. (Contributed by Scott Fenton, 3-May-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  R  =  ( C  \  ran  ( ( _V  (x)  _E  )(++) ( S  (x)  _V ) ) )   &    |-  A C B   &    |-  ( x  e.  X  <->  x S A )   =>    |-  ( A R B  <->  B  =  X )
 
Theoremrelbigcup 29112 The  Bigcup relationship is a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  Rel  Bigcup
 
Theorembrbigcup 29113 Binary relationship over 
Bigcup. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  B  e.  _V   =>    |-  ( A Bigcup B  <->  U. A  =  B )
 
Theoremdfbigcup2 29114  Bigcup using maps-to notation. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Bigcup  =  ( x  e.  _V  |->  U. x )
 
Theoremfobigcup 29115  Bigcup maps the universe onto itself. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Bigcup : _V -onto-> _V
 
Theoremfnbigcup 29116  Bigcup is a function over the universal class. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  Bigcup  Fn  _V
 
Theoremfvbigcup 29117 For sets,  Bigcup yields union. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  A  e.  _V   =>    |-  ( Bigcup `  A )  =  U. A
 
Theoremelfix 29118 Membership in the fixpoints of a class. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  A  e.  _V   =>    |-  ( A  e.  Fix R  <->  A R A )
 
Theoremelfix2 29119 Alternative membership in the fixpoint of a class. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  Rel  R   =>    |-  ( A  e.  Fix R  <->  A R A )
 
Theoremdffix2 29120 The fixpoints of a class in terms of its range. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Fix A  =  ran  ( A  i^i  _I  )
 
Theoremfixssdm 29121 The fixpoints of a class are a subset of its domain. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Fix A 
 C_  dom  A
 
Theoremfixssrn 29122 The fixpoints of a class are a subset of its range. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Fix A 
 C_  ran  A
 
Theoremfixcnv 29123 The fixpoints of a class are the same as those of its converse. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Fix A  =  Fix `' A
 
Theoremfixun 29124 The fixpoint operator distributes over union. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Fix ( A  u.  B )  =  ( Fix A  u.  Fix B )
 
Theoremellimits 29125 Membership in the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  A  e.  _V   =>    |-  ( A  e.  Limits  <->  Lim  A )
 
Theoremlimitssson 29126 The class of all limit ordinals is a subclass of the class of all ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  Limits  C_  On
 
Theoremdfom5b 29127 A quantifier-free definition of 
om that does not depend on ax-inf 8046. (Note: label was changed from dfom5 8058 to dfom5b 29127 to prevent naming conflict. NM 12-Feb-2013) (Contributed by Scott Fenton, 11-Apr-2012.)
 |-  om  =  ( On  i^i  |^| Limits )
 
Theoremsscoid 29128 A condition for subset and composition with identity. (Contributed by Scott Fenton, 13-Apr-2018.)
 |-  ( A  C_  (  _I  o.  B )  <->  ( Rel  A  /\  A  C_  B )
 )
 
Theoremdffun10 29129 Another potential definition of functionhood. Based on statements in http://people.math.gatech.edu/~belinfan/research/autoreas/otter/sum/fs/. (Contributed by Scott Fenton, 30-Aug-2017.)
 |-  ( Fun  F  <->  F  C_  (  _I 
 o.  ( _V  \  (
 ( _V  \  _I  )  o.  F ) ) ) )
 
Theoremelfuns 29130 Membership in the class of all functions. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  F  e.  _V   =>    |-  ( F  e.  Funs  <->  Fun  F )
 
Theoremelfunsg 29131 Closed form of elfuns 29130. (Contributed by Scott Fenton, 2-May-2014.)
 |-  ( F  e.  V  ->  ( F  e.  Funs  <->  Fun  F ) )
 
Theorembrsingle 29132 The binary relationship form of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ASingleton B  <->  B  =  { A } )
 
Theoremelsingles 29133* Membership in the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)
 |-  ( A  e.  Singletons 
 <-> 
 E. x  A  =  { x } )
 
Theoremfnsingle 29134 The singleton relationship is a function over the universe. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |- Singleton  Fn  _V
 
Theoremfvsingle 29135 The value of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Revised by Scott Fenton, 13-Apr-2018.)
 |-  (Singleton `  A )  =  { A }
 
Theoremdfsingles2 29136* Alternate definition of the class of all singletons. (Contributed by Scott Fenton, 20-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  Singletons  =  { x  |  E. y  x  =  { y } }
 
Theoremsnelsingles 29137 A singleton is a member of the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)
 |-  A  e.  _V   =>    |- 
 { A }  e.  Singletons
 
Theoremdfiota3 29138 A definiton of iota using minimal quantifiers. (Contributed by Scott Fenton, 19-Feb-2013.)
 |-  ( iota x ph )  = 
 U. U. ( { { x  |  ph } }  i^i 
 Singletons )
 
Theoremdffv5 29139 Another quantifier free definition of function value. (Contributed by Scott Fenton, 19-Feb-2013.)
 |-  ( F `  A )  = 
 U. U. ( { ( F " { A }
 ) }  i^i  Singletons )
 
Theoremunisnif 29140 Express union of singleton in terms of  if. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  U. { A }  =  if ( A  e.  _V ,  A ,  (/) )
 
Theorembrimage 29141 Binary relationship form of the Image functor. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( AImage R B  <->  B  =  ( R " A ) )
 
Theorembrimageg 29142 Closed form of brimage 29141. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( AImage R B  <->  B  =  ( R " A ) ) )
 
Theoremfunimage 29143 Image A is a function. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  Fun Image A
 
Theoremfnimage 29144* Image R is a function over the set-like portion of  R. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |- Image R  Fn  { x  |  ( R
 " x )  e. 
 _V }
 
Theoremimageval 29145* The image functor in maps-to notation. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |- Image R  =  ( x  e.  _V  |->  ( R " x ) )
 
Theoremfvimage 29146 The value of the image functor. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( A  e.  V  /\  ( R " A )  e.  W )  ->  (Image R `  A )  =  ( R " A ) )
 
Theorembrcart 29147 Binary relationship form of the cartesian product operator. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >.Cart C  <->  C  =  ( A  X.  B ) )
 
Theorembrdomain 29148 The binary relationship form of the domain function. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ADomain B  <->  B  =  dom  A )
 
Theorembrrange 29149 The binary relationship form of the range function. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ARange B  <->  B  =  ran  A )
 
Theorembrdomaing 29150 Closed form of brdomain 29148. (Contributed by Scott Fenton, 2-May-2014.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( ADomain B  <->  B  =  dom  A ) )
 
Theorembrrangeg 29151 Closed form of brrange 29149. (Contributed by Scott Fenton, 3-May-2014.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( ARange B  <->  B  =  ran  A ) )
 
Theorembrimg 29152 The binary relationship form of the Img function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >.Img C  <->  C  =  ( A " B ) )
 
Theorembrapply 29153 The binary relationship form of the Apply function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >.Apply C  <->  C  =  ( A `  B ) )
 
Theorembrcup 29154 Binary relationship form of the Cup function. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >.Cup C  <->  C  =  ( A  u.  B ) )
 
Theorembrcap 29155 Binary relationship form of the Cap function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >.Cap C  <->  C  =  ( A  i^i  B ) )
 
Theorembrsuccf 29156 Binary relationship form of the Succ function. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ASucc B  <->  B  =  suc  A )
 
Theoremfunpartlem 29157* Lemma for funpartfun 29158. Show membership in the restriction. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  ( A  e.  dom  ( (Image
 F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  <->  E. x ( F
 " { A }
 )  =  { x } )
 
Theoremfunpartfun 29158 The functional part of  F is a function. (Contributed by Scott Fenton, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  Fun Funpart F
 
Theoremfunpartss 29159 The functional part of  F is a subset of  F. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |- Funpart F  C_  F
 
Theoremfunpartfv 29160 The function value of the functional part is identical to the original functional value. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (Funpart F `
  A )  =  ( F `  A )
 
Theoremfullfunfnv 29161 The full functional part of  F is a function over  _V. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |- FullFun F  Fn  _V
 
Theoremfullfunfv 29162 The function value of the full function of  F agrees with  F. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (FullFun F `
  A )  =  ( F `  A )
 
Theorembrfullfun 29163 A binary relationship form condition for the full function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( AFullFun F B  <->  B  =  ( F `  A ) )
 
Theorembrrestrict 29164 The binary relationship form of the Restrict function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >.Restrict
 C 
 <->  C  =  ( A  |`  B ) )
 
Theoremdfrdg4 29165 A quantifier-free definition of the recursive definition generator. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  rec ( F ,  A )  =  U. ( (
 Funs  i^i  ( `'Domain " On ) )  \  dom  (
 ( `'  _E  o. Domain ) 
 \  Fix ( `'Apply  o.  (
 ( ( _V  X.  { (/) } )  X.  { U. { A } }
 )  u.  ( ( ( Bigcup  o. Img )  |`  ( _V 
 X.  Limits ) )  u.  ( (FullFun F  o.  (Apply  o. pprod (  _I  ,  Bigcup ) ) )  |`  ( _V  X.  ran Succ ) ) ) ) ) ) )
 
Theoremtfrqfree 29166* Calculate a quantifier-free version of the function from tfr1 7058 through tfr3 7060. (Contributed by Scott Fenton, 29-Apr-2014.)
 |-  (
 ( Funs  i^i  ( `'Domain " On ) )  \  dom  ( ( `'  _E  o. Domain )  \  Fix ( `'Apply  o.  (FullFun G  o. Restrict ) ) ) )  =  { f  |  E. x  e.  On  (
 f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }
 
Theoremdfint3 29167 Quantifier-free definition of class intersection. (Contributed by Scott Fenton, 13-Apr-2018.)
 |-  |^| A  =  ( _V  \  ( `' ( _V  \  _E  ) " A ) )
 
Theoremimagesset 29168 The Image functor applied to the converse of the subset relationship yields a subset of the subset relationship. (Contributed by Scott Fenton, 14-Apr-2018.)
 |- Image `' SSet  C_ 
 SSet
 
Theorembrub 29169* Binary relationship form of the upper bound functor. (Contributed by Scott Fenton, 3-May-2018.)
 |-  S  e.  _V   &    |-  A  e.  _V   =>    |-  ( SUB R A  <->  A. x  e.  S  x R A )
 
Theorembrlb 29170* Binary relationship form of the lower bound functor. (Contributed by Scott Fenton, 3-May-2018.)
 |-  S  e.  _V   &    |-  A  e.  _V   =>    |-  ( SLB R A  <->  A. x  e.  S  A R x )
 
21.8.39  Alternate ordered pairs
 
Syntaxcaltop 29171 Declare the syntax for an alternate ordered pair.
 class  << A ,  B >>
 
Syntaxcaltxp 29172 Declare the syntax for an alternate Cartesian product.
 class  ( A 
 XX.  B )
 
Definitiondf-altop 29173 An alternative definition of ordered pairs. This definition removes a hypothesis from its defining theorem (see altopth 29184), making it more convenient in some circumstances. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  << A ,  B >>  =  { { A } ,  { A ,  { B } } }
 
Definitiondf-altxp 29174* Define Cartesian products of alternative ordered pairs. (Contributed by Scott Fenton, 23-Mar-2012.)
 |-  ( A  XX.  B )  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  << x ,  y >> }
 
Theoremaltopex 29175 Alternative ordered pairs always exist. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  << A ,  B >>  e.  _V
 
Theoremaltopthsn 29176 Two alternate ordered pairs are equal iff the singletons of their respective elements are equal. Note that this holds regardless of sethood of any of the elements. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  ( <<
 A ,  B >>  = 
 << C ,  D >>  <->  ( { A }  =  { C }  /\  { B }  =  { D } )
 )
 
Theoremaltopeq12 29177 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  (
 ( A  =  B  /\  C  =  D ) 
 ->  << A ,  C >> 
 =  << B ,  D >> )
 
Theoremaltopeq1 29178 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  ( A  =  B  ->  << A ,  C >>  =  << B ,  C >> )
 
Theoremaltopeq2 29179 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  ( A  =  B  ->  << C ,  A >>  =  << C ,  B >> )
 
Theoremaltopth1 29180 Equality of the first members of equal alternate ordered pairs, which holds regardless of the second members' sethood. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  ( A  e.  V  ->  (
 << A ,  B >>  = 
 << C ,  D >>  ->  A  =  C )
 )
 
Theoremaltopth2 29181 Equality of the second members of equal alternate ordered pairs, which holds regardless of the first members' sethood. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  ( B  e.  V  ->  (
 << A ,  B >>  = 
 << C ,  D >>  ->  B  =  D )
 )
 
Theoremaltopthg 29182 Alternate ordered pair theorem. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( << A ,  B >> 
 =  << C ,  D >>  <->  ( A  =  C  /\  B  =  D )
 ) )
 
Theoremaltopthbg 29183 Alternate ordered pair theorem. (Contributed by Scott Fenton, 14-Apr-2012.)
 |-  (
 ( A  e.  V  /\  D  e.  W ) 
 ->  ( << A ,  B >> 
 =  << C ,  D >>  <->  ( A  =  C  /\  B  =  D )
 ) )
 
Theoremaltopth 29184 The alternate ordered pair theorem. If two alternate ordered pairs are equal, their first elements are equal and their second elements are equal. Note that  C and  D are not required to be a set due to a peculiarity of our specific ordered pair definition, as opposed to the regular ordered pairs used here, which (as in opth 4716), requires  D to be a set. (Contributed by Scott Fenton, 23-Mar-2012.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( <<
 A ,  B >>  = 
 << C ,  D >>  <->  ( A  =  C  /\  B  =  D ) )
 
Theoremaltopthb 29185 Alternate ordered pair theorem with different sethood requirements. See altopth 29184 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)
 |-  A  e.  _V   &    |-  D  e.  _V   =>    |-  ( <<
 A ,  B >>  = 
 << C ,  D >>  <->  ( A  =  C  /\  B  =  D ) )
 
Theoremaltopthc 29186 Alternate ordered pair theorem with different sethood requirements. See altopth 29184 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)
 |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <<
 A ,  B >>  = 
 << C ,  D >>  <->  ( A  =  C  /\  B  =  D ) )
 
Theoremaltopthd 29187 Alternate ordered pair theorem with different sethood requirements. See altopth 29184 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)
 |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( <<
 A ,  B >>  = 
 << C ,  D >>  <->  ( A  =  C  /\  B  =  D ) )
 
Theoremaltxpeq1 29188 Equality for alternate Cartesian products. (Contributed by Scott Fenton, 24-Mar-2012.)
 |-  ( A  =  B  ->  ( A  XX.  C )  =  ( B  XX.  C ) )
 
Theoremaltxpeq2 29189 Equality for alternate Cartesian products. (Contributed by Scott Fenton, 24-Mar-2012.)
 |-  ( A  =  B  ->  ( C  XX.  A )  =  ( C  XX.  B ) )
 
Theoremelaltxp 29190* Membership in alternate Cartesian products. (Contributed by Scott Fenton, 23-Mar-2012.)
 |-  ( X  e.  ( A  XX. 
 B )  <->  E. x  e.  A  E. y  e.  B  X  =  << x ,  y >> )
 
Theoremaltopelaltxp 29191 Alternate ordered pair membership in a Cartesian product. Note that, unlike opelxp 5023, there is no sethood requirement here. (Contributed by Scott Fenton, 22-Mar-2012.)
 |-  ( <<
 X ,  Y >>  e.  ( A  XX.  B )  <-> 
 ( X  e.  A  /\  Y  e.  B ) )
 
Theoremaltxpsspw 29192 An inclusion rule for alternate Cartesian products. (Contributed by Scott Fenton, 24-Mar-2012.)
 |-  ( A  XX.  B )  C_  ~P
 ~P ( A  u.  ~P B )
 
Theoremaltxpexg 29193 The alternate Cartesian product of two sets is a set. (Contributed by Scott Fenton, 24-Mar-2012.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( A  XX.  B )  e.  _V )
 
Theoremrankaltopb 29194 Compute the rank of an alternate ordered pair. (Contributed by Scott Fenton, 18-Dec-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( A  e.  U. ( R1 " On )  /\  B  e.  U. ( R1 " On ) ) 
 ->  ( rank `  << A ,  B >> )  =  suc  suc  ( ( rank `  A )  u.  suc  ( rank `  B ) ) )
 
Theoremnfaltop 29195 Bound-variable hypothesis builder for alternate ordered pairs. (Contributed by Scott Fenton, 25-Sep-2015.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x << A ,  B >>
 
Theoremsbcaltop 29196* Distribution of class substitution over alternate ordered pairs. (Contributed by Scott Fenton, 25-Sep-2015.)
 |-  ( A  e.  _V  ->  [_ A  /  x ]_ << C ,  D >>  =  << [_ A  /  x ]_ C ,  [_ A  /  x ]_ D >> )
 
21.8.40  Geometry in the Euclidean space
 
21.8.40.1  Congruence properties
 
Syntaxcofs 29197 Declare the syntax for the outer five segment configuration.
 class  OuterFiveSeg
 
Definitiondf-ofs 29198* The outer five segment configuration is an abbreviation for the conditions of the Five Segment Axiom (ax5seg 23912). See brofs 29220 and 5segofs 29221 for how it is used. Definition 2.10 of [Schwabhauser] p. 28. (Contributed by Scott Fenton, 21-Sep-2013.)
 |-  OuterFiveSeg  =  { <. p ,  q >.  |  E. n  e.  NN  E. a  e.  ( EE `  n ) E. b  e.  ( EE `  n ) E. c  e.  ( EE `  n ) E. d  e.  ( EE `  n ) E. x  e.  ( EE `  n ) E. y  e.  ( EE `  n ) E. z  e.  ( EE `  n ) E. w  e.  ( EE `  n ) ( p  =  <. <. a ,  b >. ,  <. c ,  d >. >.  /\  q  =  <.
 <. x ,  y >. , 
 <. z ,  w >. >.  /\  ( ( b  Btwn  <.
 a ,  c >.  /\  y  Btwn  <. x ,  z >. )  /\  ( <. a ,  b >.Cgr <. x ,  y >.  /\ 
 <. b ,  c >.Cgr <.
 y ,  z >. ) 
 /\  ( <. a ,  d >.Cgr <. x ,  w >.  /\  <. b ,  d >.Cgr
 <. y ,  w >. ) ) ) }
 
Theoremcgrrflx2d 29199 Deduction form of axcgrrflx 23888. (Contributed by Scott Fenton, 13-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   =>    |-  ( ph  ->  <. A ,  B >.Cgr <. B ,  A >. )
 
Theoremcgrtr4d 29200 Deduction form of axcgrtr 23889. (Contributed by Scott Fenton, 13-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   &    |-  ( ph  ->  C  e.  ( EE `  N ) )   &    |-  ( ph  ->  D  e.  ( EE `  N ) )   &    |-  ( ph  ->  E  e.  ( EE `  N ) )   &    |-  ( ph  ->  F  e.  ( EE `  N ) )   &    |-  ( ph  ->  <. A ,  B >.Cgr <. C ,  D >. )   &    |-  ( ph  ->  <. A ,  B >.Cgr <. E ,  F >. )   =>    |-  ( ph  ->  <. C ,  D >.Cgr <. E ,  F >. )
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