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Theorem List for Metamath Proof Explorer - 8101-8200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremaxpowndlem3 8101* Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.) (Revised by Mario Carneiro, 10-Dec-2016.)
 |-  ( -.  x  =  y  ->  E. x A. y ( A. x ( E. z  x  e.  y  ->  A. y  x  e.  z )  ->  y  e.  x )
 )
 
Theoremaxpowndlem4 8102 Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)
 |-  ( -.  A. y  y  =  x  ->  ( -.  A. y  y  =  z  ->  ( -.  x  =  y  ->  E. x A. y
 ( A. x ( E. z  x  e.  y  ->  A. y  x  e.  z )  ->  y  e.  x ) ) ) )
 
Theoremaxpownd 8103 A version of the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.)
 |-  ( -.  x  =  y  ->  E. x A. y ( A. x ( E. z  x  e.  y  ->  A. y  x  e.  z )  ->  y  e.  x )
 )
 
Theoremaxregndlem1 8104 Lemma for the Axiom of Regularity with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)
 |-  ( A. x  x  =  z  ->  ( x  e.  y  ->  E. x ( x  e.  y  /\  A. z
 ( z  e.  x  ->  -.  z  e.  y
 ) ) ) )
 
Theoremaxregndlem2 8105* Lemma for the Axiom of Regularity with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)
 |-  ( x  e.  y  ->  E. x ( x  e.  y  /\  A. z ( z  e.  x  ->  -.  z  e.  y ) ) )
 
Theoremaxregnd 8106 A version of the Axiom of Regularity with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)
 |-  ( x  e.  y  ->  E. x ( x  e.  y  /\  A. z ( z  e.  x  ->  -.  z  e.  y ) ) )
 
Theoremaxinfndlem1 8107* Lemma for the Axiom of Infinity with no distinct variable conditions. (Contributed by NM, 5-Jan-2002.)
 |-  ( A. x  y  e.  z  ->  E. x ( y  e.  x  /\  A. y ( y  e.  x  ->  E. z
 ( y  e.  z  /\  z  e.  x ) ) ) )
 
Theoremaxinfnd 8108 A version of the Axiom of Infinity with no distinct variable conditions. (Contributed by NM, 5-Jan-2002.)
 |- 
 E. x ( y  e.  z  ->  (
 y  e.  x  /\  A. y ( y  e.  x  ->  E. z
 ( y  e.  z  /\  z  e.  x ) ) ) )
 
Theoremaxacndlem1 8109 Lemma for the Axiom of Choice with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)
 |-  ( A. x  x  =  y  ->  E. x A. y A. z (
 A. x ( y  e.  z  /\  z  e.  w )  ->  E. w A. y ( E. w ( ( y  e.  z  /\  z  e.  w )  /\  (
 y  e.  w  /\  w  e.  x )
 ) 
 <->  y  =  w ) ) )
 
Theoremaxacndlem2 8110 Lemma for the Axiom of Choice with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)
 |-  ( A. x  x  =  z  ->  E. x A. y A. z (
 A. x ( y  e.  z  /\  z  e.  w )  ->  E. w A. y ( E. w ( ( y  e.  z  /\  z  e.  w )  /\  (
 y  e.  w  /\  w  e.  x )
 ) 
 <->  y  =  w ) ) )
 
Theoremaxacndlem3 8111 Lemma for the Axiom of Choice with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)
 |-  ( A. y  y  =  z  ->  E. x A. y A. z (
 A. x ( y  e.  z  /\  z  e.  w )  ->  E. w A. y ( E. w ( ( y  e.  z  /\  z  e.  w )  /\  (
 y  e.  w  /\  w  e.  x )
 ) 
 <->  y  =  w ) ) )
 
Theoremaxacndlem4 8112* Lemma for the Axiom of Choice with no distinct variable conditions. (Contributed by NM, 8-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)
 |- 
 E. x A. y A. z ( A. x ( y  e.  z  /\  z  e.  w )  ->  E. w A. y
 ( E. w ( ( y  e.  z  /\  z  e.  w )  /\  ( y  e.  w  /\  w  e.  x ) )  <->  y  =  w ) )
 
Theoremaxacndlem5 8113* Lemma for the Axiom of Choice with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)
 |- 
 E. x A. y A. z ( A. x ( y  e.  z  /\  z  e.  w )  ->  E. w A. y
 ( E. w ( ( y  e.  z  /\  z  e.  w )  /\  ( y  e.  w  /\  w  e.  x ) )  <->  y  =  w ) )
 
Theoremaxacnd 8114 A version of the Axiom of Choice with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)
 |- 
 E. x A. y A. z ( A. x ( y  e.  z  /\  z  e.  w )  ->  E. w A. y
 ( E. w ( ( y  e.  z  /\  z  e.  w )  /\  ( y  e.  w  /\  w  e.  x ) )  <->  y  =  w ) )
 
Theoremzfcndext 8115* Axiom of Extensionality ax-ext 2234, reproved from conditionless ZFC version and predicate calculus. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)
 |-  ( A. z ( z  e.  x  <->  z  e.  y
 )  ->  x  =  y )
 
Theoremzfcndrep 8116* Axiom of Replacement ax-rep 4028, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)
 |-  ( A. w E. y A. z ( A. y ph  ->  z  =  y )  ->  E. y A. z ( z  e.  y  <->  E. w ( w  e.  x  /\  A. y ph ) ) )
 
Theoremzfcndun 8117* Axiom of Union ax-un 4403, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)
 |- 
 E. y A. z
 ( E. w ( z  e.  w  /\  w  e.  x )  ->  z  e.  y )
 
Theoremzfcndpow 8118* Axiom of Power Sets ax-pow 4082, reproved from conditionless ZFC axioms. The proof uses the "Axiom of Twoness," dtru 4095. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)
 |- 
 E. y A. z
 ( A. w ( w  e.  z  ->  w  e.  x )  ->  z  e.  y )
 
Theoremzfcndreg 8119* Axiom of Regularity ax-reg 7190, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)
 |-  ( E. y  y  e.  x  ->  E. y
 ( y  e.  x  /\  A. z ( z  e.  y  ->  -.  z  e.  x ) ) )
 
Theoremzfcndinf 8120* Axiom of Infinity ax-inf 7223, reproved from conditionless ZFC axioms. Since we have already reproved Extensionality, Replacement, and Power Sets above, we are justified in referencing theorem el 4086 in the proof. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)
 |- 
 E. y ( x  e.  y  /\  A. z ( z  e.  y  ->  E. w ( z  e.  w  /\  w  e.  y
 ) ) )
 
Theoremzfcndac 8121* Axiom of Choice ax-ac 7969, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)
 |- 
 E. y A. z A. w ( ( z  e.  w  /\  w  e.  x )  ->  E. v A. u ( E. t
 ( ( u  e.  w  /\  w  e.  t )  /\  ( u  e.  t  /\  t  e.  y )
 ) 
 <->  u  =  v ) )
 
3.4  The Generalized Continuum Hypothesis
 
Syntaxcgch 8122 Extend class notation to include the collection of sets that satisfy the GCH.
 class GCH
 
Definitiondf-gch 8123* Define the collection of "GCH-sets", or sets for which the generalized continuum hypothesis holds. In this language the generalized continuum hypothesis can be expressed as GCH  =  _V. A set  x satisfies the generalized continuum hypothesis if it is finite or there is no set  y strictly between  x and its powerset in cardinality. The continuum hypothesis is equivalent to  om  e. GCH. (Contributed by Mario Carneiro, 15-May-2015.)
 |- GCH 
 =  ( Fin  u.  { x  |  A. y  -.  ( x  ~<  y  /\  y  ~<  ~P x ) }
 )
 
Theoremelgch 8124* Elementhood in the collection of GCH-sets. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( A  e.  V  ->  ( A  e. GCH  <->  ( A  e.  Fin 
 \/  A. x  -.  ( A  ~<  x  /\  x  ~<  ~P A ) ) ) )
 
Theoremfingch 8125 A finite set is a GCH-set. (Contributed by Mario Carneiro, 15-May-2015.)
 |- 
 Fin  C_ GCH
 
Theoremgchi 8126 The only GCH-sets which have other sets between it and its power set are finite sets. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ( A  e. GCH  /\  A  ~<  B  /\  B  ~<  ~P A )  ->  A  e.  Fin )
 
Theoremgchen1 8127 If  A  <_  B  <  ~P A, and  A is an infinite GCH-set, then  A  =  B in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<  ~P A ) )  ->  A  ~~  B )
 
Theoremgchen2 8128 If  A  <  B  <_  ~P A, and  A is an infinite GCH-set, then  B  =  ~P A in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A 
 ~<  B  /\  B  ~<_  ~P A ) )  ->  B  ~~  ~P A )
 
Theoremgchor 8129 If  A  <_  B  <_  ~P A, and  A is an infinite GCH-set, then either  A  =  B or  B  =  ~P A in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ( ( A  e. GCH  /\  -.  A  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  ~P A ) )  ->  ( A 
 ~~  B  \/  B  ~~ 
 ~P A ) )
 
Theoremengch 8130 The property of being a GCH-set is a cardinal invariant. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( A  ~~  B  ->  ( A  e. GCH  <->  B  e. GCH ) )
 
Theoremgchdomtri 8131 Under certain conditions, a GCH-set can demonstrate trichotomy of dominance. Lemma for gchac 8175. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ( A  e. GCH  /\  ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A )  ->  ( A 
 ~<_  B  \/  B  ~<_  A ) )
 
Theoremfpwwe2cbv 8132* Lemma for fpwwe2 8145. (Contributed by Mario Carneiro, 3-Jun-2015.)
 |-  W  =  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) ) 
 /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y ) ) }   =>    |-  W  =  { <. a ,  s >.  |  ( ( a  C_  A  /\  s  C_  ( a  X.  a ) ) 
 /\  ( s  We  a  /\  A. z  e.  a  [. ( `' s " { z } )  /  v ]. ( v F ( s  i^i  ( v  X.  v ) ) )  =  z ) ) }
 
Theoremfpwwe2lem1 8133* Lemma for fpwwe2 8145. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  W  =  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) ) 
 /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y ) ) }   =>    |-  W  C_  ( ~P A  X.  ~P ( A  X.  A ) )
 
Theoremfpwwe2lem2 8134* Lemma for fpwwe2 8145. (Contributed by Mario Carneiro, 19-May-2015.)
 |-  W  =  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) ) 
 /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y ) ) }   &    |-  ( ph  ->  A  e.  _V )   =>    |-  ( ph  ->  ( X W R  <->  ( ( X 
 C_  A  /\  R  C_  ( X  X.  X ) )  /\  ( R  We  X  /\  A. y  e.  X  [. ( `' R " { y } )  /  u ]. ( u F ( R  i^i  ( u  X.  u ) ) )  =  y ) ) ) )
 
Theoremfpwwe2lem3 8135* Lemma for fpwwe2 8145. (Contributed by Mario Carneiro, 19-May-2015.)
 |-  W  =  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) ) 
 /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y ) ) }   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  X W R )   =>    |-  ( ( ph  /\  B  e.  X )  ->  (
 ( `' R " { B } ) F ( R  i^i  (
 ( `' R " { B } )  X.  ( `' R " { B } ) ) ) )  =  B )
 
Theoremfpwwe2lem5 8136* Lemma for fpwwe2 8145. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  W  =  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) ) 
 /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y ) ) }   &    |-  ( ph  ->  A  e.  _V )   &    |-  (
 ( ph  /\  ( x 
 C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x ) )  ->  ( x F r )  e.  A )   =>    |-  ( ( ph  /\  ( X  C_  A  /\  R  C_  ( X  X.  X )  /\  R  We  X ) )  ->  ( X F R )  e.  A )
 
Theoremfpwwe2lem6 8137* Lemma for fpwwe2 8145. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  W  =  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) ) 
 /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y ) ) }   &    |-  ( ph  ->  A  e.  _V )   &    |-  (
 ( ph  /\  ( x 
 C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x ) )  ->  ( x F r )  e.  A )   &    |-  ( ph  ->  X W R )   &    |-  ( ph  ->  Y W S )   &    |-  M  = OrdIso ( R ,  X )   &    |-  N  = OrdIso ( S ,  Y )   &    |-  ( ph  ->  B  e.  dom  M )   &    |-  ( ph  ->  B  e.  dom  N )   &    |-  ( ph  ->  ( M  |`  B )  =  ( N  |`  B ) )   =>    |-  ( ( ph  /\  C R ( M `  B ) )  ->  ( C  e.  X  /\  C  e.  Y  /\  ( `' M `  C )  =  ( `' N `  C ) ) )
 
Theoremfpwwe2lem7 8138* Lemma for fpwwe2 8145. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  W  =  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) ) 
 /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y ) ) }   &    |-  ( ph  ->  A  e.  _V )   &    |-  (
 ( ph  /\  ( x 
 C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x ) )  ->  ( x F r )  e.  A )   &    |-  ( ph  ->  X W R )   &    |-  ( ph  ->  Y W S )   &    |-  M  = OrdIso ( R ,  X )   &    |-  N  = OrdIso ( S ,  Y )   &    |-  ( ph  ->  B  e.  dom  M )   &    |-  ( ph  ->  B  e.  dom  N )   &    |-  ( ph  ->  ( M  |`  B )  =  ( N  |`  B ) )   =>    |-  ( ( ph  /\  C R ( M `  B ) )  ->  ( C S ( N `
  B )  /\  ( D R ( M `
  B )  ->  ( C R D  <->  C S D ) ) ) )
 
Theoremfpwwe2lem8 8139* Lemma for fpwwe2 8145. Show by induction that the two isometries  M and  N agree on their common domain. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  W  =  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) ) 
 /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y ) ) }   &    |-  ( ph  ->  A  e.  _V )   &    |-  (
 ( ph  /\  ( x 
 C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x ) )  ->  ( x F r )  e.  A )   &    |-  ( ph  ->  X W R )   &    |-  ( ph  ->  Y W S )   &    |-  M  = OrdIso ( R ,  X )   &    |-  N  = OrdIso ( S ,  Y )   &    |-  ( ph  ->  dom  M  C_ 
 dom  N )   =>    |-  ( ph  ->  M  =  ( N  |`  dom  M ) )
 
Theoremfpwwe2lem9 8140* Lemma for fpwwe2 8145. Given two well-orders  <. X ,  R >. and  <. Y ,  S >. of parts of  A, one is an initial segment of the other. (The  O  C_  P hypothesis is in order to break the symmetry of  X and  Y.) (Contributed by Mario Carneiro, 15-May-2015.)
 |-  W  =  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) ) 
 /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y ) ) }   &    |-  ( ph  ->  A  e.  _V )   &    |-  (
 ( ph  /\  ( x 
 C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x ) )  ->  ( x F r )  e.  A )   &    |-  ( ph  ->  X W R )   &    |-  ( ph  ->  Y W S )   &    |-  M  = OrdIso ( R ,  X )   &    |-  N  = OrdIso ( S ,  Y )   &    |-  ( ph  ->  dom  M  C_ 
 dom  N )   =>    |-  ( ph  ->  ( X  C_  Y  /\  R  =  ( S  i^i  ( Y  X.  X ) ) ) )
 
Theoremfpwwe2lem10 8141* Lemma for fpwwe2 8145. Given two well-orders  <. X ,  R >. and  <. Y ,  S >. of parts of  A, one is an initial segment of the other. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  W  =  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) ) 
 /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y ) ) }   &    |-  ( ph  ->  A  e.  _V )   &    |-  (
 ( ph  /\  ( x 
 C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x ) )  ->  ( x F r )  e.  A )   &    |-  ( ph  ->  X W R )   &    |-  ( ph  ->  Y W S )   =>    |-  ( ph  ->  (
 ( X  C_  Y  /\  R  =  ( S  i^i  ( Y  X.  X ) ) )  \/  ( Y  C_  X  /\  S  =  ( R  i^i  ( X  X.  Y ) ) ) ) )
 
Theoremfpwwe2lem11 8142* Lemma for fpwwe2 8145. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  W  =  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) ) 
 /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y ) ) }   &    |-  ( ph  ->  A  e.  _V )   &    |-  (
 ( ph  /\  ( x 
 C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x ) )  ->  ( x F r )  e.  A )   &    |-  X  =  U. dom  W   =>    |-  ( ph  ->  W : dom  W --> ~P ( X  X.  X ) )
 
Theoremfpwwe2lem12 8143* Lemma for fpwwe2 8145. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  W  =  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) ) 
 /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y ) ) }   &    |-  ( ph  ->  A  e.  _V )   &    |-  (
 ( ph  /\  ( x 
 C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x ) )  ->  ( x F r )  e.  A )   &    |-  X  =  U. dom  W   =>    |-  ( ph  ->  X  e.  dom  W )
 
Theoremfpwwe2lem13 8144* Lemma for fpwwe2 8145. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  W  =  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) ) 
 /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y ) ) }   &    |-  ( ph  ->  A  e.  _V )   &    |-  (
 ( ph  /\  ( x 
 C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x ) )  ->  ( x F r )  e.  A )   &    |-  X  =  U. dom  W   =>    |-  ( ph  ->  ( X F ( W `  X ) )  e.  X )
 
Theoremfpwwe2 8145* Given any function  F from well-orderings of subsets of 
A to  A, there is a unique well-ordered subset  <. X ,  ( W `  X )
>. which "agrees" with  F in the sense that each initial segment maps to its upper bound, and such that the entire set maps to an element of the set (so that it cannot be extended without losing the well-ordering). This theorem can be used to prove dfac8a 7541. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  W  =  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) ) 
 /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y ) ) }   &    |-  ( ph  ->  A  e.  _V )   &    |-  (
 ( ph  /\  ( x 
 C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x ) )  ->  ( x F r )  e.  A )   &    |-  X  =  U. dom  W   =>    |-  ( ph  ->  (
 ( Y W R  /\  ( Y F R )  e.  Y )  <->  ( Y  =  X  /\  R  =  ( W `  X ) ) ) )
 
Theoremfpwwecbv 8146* Lemma for fpwwe 8148. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  W  =  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) ) 
 /\  ( r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } )
 )  =  y ) ) }   =>    |-  W  =  { <. a ,  s >.  |  ( ( a  C_  A  /\  s  C_  ( a  X.  a ) ) 
 /\  ( s  We  a  /\  A. z  e.  a  ( F `  ( `' s " { z } )
 )  =  z ) ) }
 
Theoremfpwwelem 8147* Lemma for fpwwe 8148. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  W  =  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) ) 
 /\  ( r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } )
 )  =  y ) ) }   &    |-  ( ph  ->  A  e.  _V )   =>    |-  ( ph  ->  ( X W R  <->  ( ( X 
 C_  A  /\  R  C_  ( X  X.  X ) )  /\  ( R  We  X  /\  A. y  e.  X  ( F `  ( `' R " { y } )
 )  =  y ) ) ) )
 
Theoremfpwwe 8148* Given any function  F from the powerset of  A to  A, canth2 6899 gives that the function is not injective, but we can say rather more than that. There is a unique well-ordered subset  <. X , 
( W `  X
) >. which "agrees" with  F in the sense that each initial segment maps to its upper bound, and such that the entire set maps to an element of the set (so that it cannot be extended without losing the well-ordering). This theorem can be used to prove dfac8a 7541. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  W  =  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) ) 
 /\  ( r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } )
 )  =  y ) ) }   &    |-  ( ph  ->  A  e.  _V )   &    |-  (
 ( ph  /\  x  e.  ( ~P A  i^i  dom  card ) )  ->  ( F `  x )  e.  A )   &    |-  X  =  U. dom  W   =>    |-  ( ph  ->  (
 ( Y W R  /\  ( F `  Y )  e.  Y )  <->  ( Y  =  X  /\  R  =  ( W `  X ) ) ) )
 
Theoremcanth4 8149* An "effective" form of Cantor's theorem canth 6178. For any function  F from the powerset of  A to  A, there are two definable sets  B and  C which witness non-injectivity of  F. Corollary 1.3 of [KanamoriPincus] p. 416. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  W  =  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) ) 
 /\  ( r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } )
 )  =  y ) ) }   &    |-  B  =  U. dom  W   &    |-  C  =  ( `' ( W `  B ) " { ( F `
  B ) }
 )   =>    |-  ( ( A  e.  V  /\  F : D --> A  /\  ( ~P A  i^i  dom  card )  C_  D )  ->  ( B  C_  A  /\  C  C.  B  /\  ( F `  B )  =  ( F `  C ) ) )
 
Theoremcanthnumlem 8150* Lemma for canthnum 8151. (Contributed by Mario Carneiro, 19-May-2015.)
 |-  W  =  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) ) 
 /\  ( r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } )
 )  =  y ) ) }   &    |-  B  =  U. dom  W   &    |-  C  =  ( `' ( W `  B ) " { ( F `
  B ) }
 )   =>    |-  ( A  e.  V  ->  -.  F : ( ~P A  i^i  dom  card
 ) -1-1-> A )
 
Theoremcanthnum 8151 The set of well-orderable subsets of a set  A strictly dominates  A. A stronger form of canth2 6899. Corollary 1.4(a) of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 19-May-2015.)
 |-  ( A  e.  V  ->  A  ~<  ( ~P A  i^i  dom  card ) )
 
Theoremcanthwelem 8152* Lemma for canthnum 8151. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  O  =  { <. x ,  r >.  |  ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x ) }   &    |-  W  =  { <. x ,  r >.  |  ( ( x 
 C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y ) ) }   &    |-  B  =  U. dom  W   &    |-  C  =  ( `' ( W `  B ) " { ( B F ( W `  B ) ) }
 )   =>    |-  ( A  e.  V  ->  -.  F : O -1-1-> A )
 
Theoremcanthwe 8153* The set of well-orders of a set  A strictly dominates  A. A stronger form of canth2 6899. Corollary 1.4(b) of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  O  =  { <. x ,  r >.  |  ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x ) }   =>    |-  ( A  e.  V  ->  A  ~<  O )
 
Theoremcanthp1lem1 8154 Lemma for canthp1 8156. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( 1o  ~<  A  ->  ( A  +c  2o )  ~<_  ~P A )
 
Theoremcanthp1lem2 8155* Lemma for canthp1 8156. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( ph  ->  1o  ~<  A )   &    |-  ( ph  ->  F : ~P A -1-1-onto-> ( A  +c  1o ) )   &    |-  ( ph  ->  G : ( ( A  +c  1o )  \  { ( F `  A ) } ) -1-1-onto-> A )   &    |-  H  =  ( ( G  o.  F )  o.  ( x  e. 
 ~P A  |->  if ( x  =  A ,  (/)
 ,  x ) ) )   &    |-  W  =  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) ) 
 /\  ( r  We  x  /\  A. y  e.  x  ( H `  ( `' r " { y } )
 )  =  y ) ) }   &    |-  B  =  U. dom  W   =>    |- 
 -.  ph
 
Theoremcanthp1 8156 A slightly stronger form of Cantor's theorem: For  1  <  n,  n  +  1  <  2 ^ n. Corollary 1.6 of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( 1o  ~<  A  ->  ( A  +c  1o )  ~<  ~P A )
 
Theoremfinngch 8157 The exclusion of finite sets from consideration in df-gch 8123 is necessary, because otherwise finite sets larger than a singleton would violate the GCH property. (Contributed by Mario Carneiro, 10-Jun-2015.)
 |-  ( ( A  e.  Fin  /\  1o  ~<  A )  ->  ( A  ~<  ( A  +c  1o )  /\  ( A  +c  1o )  ~<  ~P A ) )
 
Theoremgchcda1 8158 An infinite GCH-set is idempotent under cardinal successor. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( ( A  e. GCH  /\ 
 -.  A  e.  Fin )  ->  ( A  +c  1o )  ~~  A )
 
Theoremgchinf 8159 An infinite GCH-set is Dedekind-infinite. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  ( ( A  e. GCH  /\ 
 -.  A  e.  Fin )  ->  om  ~<_  A )
 
Theorempwfseqlem1 8160* Lemma for pwfseq 8166. Derive a contradiction by diagonalization. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  ( ph  ->  G : ~P A -1-1-> U_ n  e.  om  ( A  ^m  n ) )   &    |-  ( ph  ->  X  C_  A )   &    |-  ( ph  ->  H : om
 -1-1-onto-> X )   &    |-  ( ps  <->  ( ( x 
 C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  /\  om  ~<_  x ) )   &    |-  ( ( ph  /\ 
 ps )  ->  K : U_ n  e.  om  ( x  ^m  n )
 -1-1-> x )   &    |-  D  =  ( G `  { w  e.  x  |  (
 ( `' K `  w )  e.  ran  G 
 /\  -.  w  e.  ( `' G `  ( `' K `  w ) ) ) } )   =>    |-  (
 ( ph  /\  ps )  ->  D  e.  ( U_ n  e.  om  ( A 
 ^m  n )  \  U_ n  e.  om  ( x  ^m  n ) ) )
 
Theorempwfseqlem2 8161* Lemma for pwfseq 8166. (Contributed by Mario Carneiro, 18-Nov-2014.)
 |-  ( ph  ->  G : ~P A -1-1-> U_ n  e.  om  ( A  ^m  n ) )   &    |-  ( ph  ->  X  C_  A )   &    |-  ( ph  ->  H : om
 -1-1-onto-> X )   &    |-  ( ps  <->  ( ( x 
 C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  /\  om  ~<_  x ) )   &    |-  ( ( ph  /\ 
 ps )  ->  K : U_ n  e.  om  ( x  ^m  n )
 -1-1-> x )   &    |-  D  =  ( G `  { w  e.  x  |  (
 ( `' K `  w )  e.  ran  G 
 /\  -.  w  e.  ( `' G `  ( `' K `  w ) ) ) } )   &    |-  F  =  ( x  e.  _V ,  r  e.  _V  |->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^|
 { z  e.  om  |  -.  ( D `  z )  e.  x } ) ) )   =>    |-  ( ( Y  e.  Fin  /\  R  e.  _V )  ->  ( Y F R )  =  ( H `  ( card `  Y )
 ) )
 
Theorempwfseqlem3 8162* Lemma for pwfseq 8166. Using the construction  D from pwfseqlem1 8160, produce a function  F that maps any well-ordered infinite set to an element outside the set. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  ( ph  ->  G : ~P A -1-1-> U_ n  e.  om  ( A  ^m  n ) )   &    |-  ( ph  ->  X  C_  A )   &    |-  ( ph  ->  H : om
 -1-1-onto-> X )   &    |-  ( ps  <->  ( ( x 
 C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  /\  om  ~<_  x ) )   &    |-  ( ( ph  /\ 
 ps )  ->  K : U_ n  e.  om  ( x  ^m  n )
 -1-1-> x )   &    |-  D  =  ( G `  { w  e.  x  |  (
 ( `' K `  w )  e.  ran  G 
 /\  -.  w  e.  ( `' G `  ( `' K `  w ) ) ) } )   &    |-  F  =  ( x  e.  _V ,  r  e.  _V  |->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^|
 { z  e.  om  |  -.  ( D `  z )  e.  x } ) ) )   =>    |-  ( ( ph  /\  ps )  ->  ( x F r )  e.  ( A  \  x ) )
 
Theorempwfseqlem4a 8163* Lemma for pwfseqlem4 8164. (Contributed by Mario Carneiro, 7-Jun-2016.)
 |-  ( ph  ->  G : ~P A -1-1-> U_ n  e.  om  ( A  ^m  n ) )   &    |-  ( ph  ->  X  C_  A )   &    |-  ( ph  ->  H : om
 -1-1-onto-> X )   &    |-  ( ps  <->  ( ( x 
 C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  /\  om  ~<_  x ) )   &    |-  ( ( ph  /\ 
 ps )  ->  K : U_ n  e.  om  ( x  ^m  n )
 -1-1-> x )   &    |-  D  =  ( G `  { w  e.  x  |  (
 ( `' K `  w )  e.  ran  G 
 /\  -.  w  e.  ( `' G `  ( `' K `  w ) ) ) } )   &    |-  F  =  ( x  e.  _V ,  r  e.  _V  |->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^|
 { z  e.  om  |  -.  ( D `  z )  e.  x } ) ) )   =>    |-  ( ( ph  /\  (
 a  C_  A  /\  s  C_  ( a  X.  a )  /\  s  We  a ) )  ->  ( a F s )  e.  A )
 
Theorempwfseqlem4 8164* Lemma for pwfseq 8166. Derive a final contradiction from the function  F in pwfseqlem3 8162. Applying fpwwe2 8145 to it, we get a certain maximal well-ordered subset 
Z, but the defining property  ( Z F ( W `  Z
) )  e.  Z contradicts our assumption on  F, so we are reduced to the case of 
Z finite. This too is a contradiction, though, because  Z and its preimage under  ( W `  Z
) are distinct sets of the same cardinality and in a subset relation, which is impossible for finite sets. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  ( ph  ->  G : ~P A -1-1-> U_ n  e.  om  ( A  ^m  n ) )   &    |-  ( ph  ->  X  C_  A )   &    |-  ( ph  ->  H : om
 -1-1-onto-> X )   &    |-  ( ps  <->  ( ( x 
 C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  /\  om  ~<_  x ) )   &    |-  ( ( ph  /\ 
 ps )  ->  K : U_ n  e.  om  ( x  ^m  n )
 -1-1-> x )   &    |-  D  =  ( G `  { w  e.  x  |  (
 ( `' K `  w )  e.  ran  G 
 /\  -.  w  e.  ( `' G `  ( `' K `  w ) ) ) } )   &    |-  F  =  ( x  e.  _V ,  r  e.  _V  |->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^|
 { z  e.  om  |  -.  ( D `  z )  e.  x } ) ) )   &    |-  W  =  { <. a ,  s >.  |  (
 ( a  C_  A  /\  s  C_  ( a  X.  a ) ) 
 /\  ( s  We  a  /\  A. b  e.  a  [. ( `' s " { b } )  /  v ]. ( v F ( s  i^i  ( v  X.  v ) ) )  =  b ) ) }   &    |-  Z  =  U. dom  W   =>    |- 
 -.  ph
 
Theorempwfseqlem5 8165* Lemma for pwfseq 8166. Although in some ways pwfseqlem4 8164 is the "main" part of the proof, one last aspect which makes up a remark in the original text is by far the hardest part to formalize. The main proof relies on the existence of an injection  K from the set of finite sequences on an infinite set 
x to  x. Now this alone would not be difficult to prove; this is mostly the claim of fseqen 7538. However, what is needed for the proof is a canonical injection on these sets, so we have to start from scratch pulling together explicit bijections from the lemmas.

If one attempts such a program, it will mostly go through, but there is one key step which is inherently nonconstructive, namely the proof of infxpen 7526. The resolution is not obvious, but it turns out that reversing an infinite ordinal's Cantor normal form absorbs all the non-leading terms (cnfcom3c 7293), which can be used to construct a pairing function explicitly using properties of the ordinal exponential (infxpenc 7529). (Contributed by Mario Carneiro, 31-May-2015.)

 |-  ( ph  ->  G : ~P A -1-1-> U_ n  e.  om  ( A  ^m  n ) )   &    |-  ( ph  ->  X  C_  A )   &    |-  ( ph  ->  H : om
 -1-1-onto-> X )   &    |-  ( ps  <->  ( ( t 
 C_  A  /\  r  C_  ( t  X.  t
 )  /\  r  We  t )  /\  om  ~<_  t ) )   &    |-  ( ph  ->  A. b  e.  (har `  ~P A ) ( om  C_  b  ->  ( N `  b ) : ( b  X.  b ) -1-1-onto-> b ) )   &    |-  O  = OrdIso (
 r ,  t )   &    |-  T  =  ( u  e.  dom  O ,  v  e.  dom  O  |->  <. ( O `
  u ) ,  ( O `  v
 ) >. )   &    |-  P  =  ( ( O  o.  ( N `  dom  O ) )  o.  `' T )   &    |-  S  = seq𝜔 ( ( k  e. 
 _V ,  f  e. 
 _V  |->  ( x  e.  ( t  ^m  suc  k )  |->  ( ( f `  ( x  |`  k ) ) P ( x `  k
 ) ) ) ) ,  { <. (/) ,  ( O `  (/) ) >. } )   &    |-  Q  =  ( y  e.  U_ n  e.  om  ( t 
 ^m  n )  |->  <. dom  y ,  ( ( S `  dom  y
 ) `  y ) >. )   &    |-  I  =  ( x  e.  om ,  y  e.  t  |->  <.
 ( O `  x ) ,  y >. )   &    |-  K  =  ( ( P  o.  I )  o.  Q )   =>    |- 
 -.  ph
 
Theorempwfseq 8166* The powerset of a Dedekind-infinite set does not inject into the set of finite sequences. The proof is due to Halbeisen and Shelah. Proposition 1.7 of [KanamoriPincus] p. 418. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  ( om  ~<_  A  ->  -. 
 ~P A  ~<_  U_ n  e.  om  ( A  ^m  n ) )
 
Theorempwxpndom2 8167 The powerset of a Dedekind-infinite set does not inject into its cross product with itself. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  ( om  ~<_  A  ->  -. 
 ~P A  ~<_  ( A  +c  ( A  X.  A ) ) )
 
Theorempwxpndom 8168 The powerset of a Dedekind-infinite set does not inject into its cross product with itself. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  ( om  ~<_  A  ->  -. 
 ~P A  ~<_  ( A  X.  A ) )
 
Theorempwcdandom 8169 The powerset of a Dedekind-infinite set does not inject into its cardinal sum with itself. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  ( om  ~<_  A  ->  -. 
 ~P A  ~<_  ( A  +c  A ) )
 
Theoremgchcdaidm 8170 An infinite GCH-set is idempotent under cardinal sum. Part of Lemma 2.2 of [KanamoriPincus] p. 419. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  ( ( A  e. GCH  /\ 
 -.  A  e.  Fin )  ->  ( A  +c  A )  ~~  A )
 
Theoremgchxpidm 8171 An infinite GCH-set is idempotent under cardinal product. Part of Lemma 2.2 of [KanamoriPincus] p. 419. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  ( ( A  e. GCH  /\ 
 -.  A  e.  Fin )  ->  ( A  X.  A )  ~~  A )
 
Theoremgchaclem 8172 Lemma for gchac 8175 (obsolete, used in Sierpiński's proof). (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ph  ->  om  ~<_  A )   &    |-  ( ph  ->  ~P C  e. GCH )   &    |-  ( ph  ->  ( A  ~<_  C  /\  ( B 
 ~<_  ~P C  ->  ~P A  ~<_  B ) ) )   =>    |-  ( ph  ->  ( A  ~<_  ~P C  /\  ( B  ~<_  ~P ~P C  ->  ~P A  ~<_  B ) ) )
 
Theoremgchhar 8173 A "local" form of gchac 8175. If  A and  ~P A are GCH-sets, then the Hartogs number of  A is  ~P A (so  ~P A and a fortiori 
A are well-orderable). The proof is due to Specker. Theorem 2.1 of [KanamoriPincus] p. 419. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  (har `  A )  ~~  ~P A )
 
Theoremgchacg 8174 A "local" form of gchac 8175. If  A and  ~P A are GCH-sets, then  ~P A is well-orderable. The proof is due to Specker. Theorem 2.1 of [KanamoriPincus] p. 419. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  ~P A  e. GCH )  ->  ~P A  e.  dom  card )
 
Theoremgchac 8175 The Generalized Continuum Hypothesis implies the Axiom of Choice. The original proof is due to Sierpiński (1947); we use a refinement of Sierpiński's result due to Specker. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  (GCH  =  _V  -> CHOICE )
 
Theoremgchpwdom 8176 A relationship between dominance over the powerset and strict dominance when the sets involved are infinite GCH-sets. Proposition 3.1 of [KanamoriPincus] p. 421. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  ( ( om  ~<_  A  /\  A  e. GCH  /\  B  e. GCH ) 
 ->  ( A  ~<  B  <->  ~P A  ~<_  B ) )
 
Theoremgchaleph 8177 If  ( aleph `  A
) is a GCH-set and its powerset is well-orderable, then the successor aleph  ( aleph `  suc  A ) is equinumerous to the powerset of  ( aleph `  A
). (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ~P ( aleph `  A )  e.  dom  card
 )  ->  ( aleph ` 
 suc  A )  ~~  ~P ( aleph `  A )
 )
 
Theoremgchaleph2 8178 If  ( aleph `  A
) and  ( aleph `  suc  A ) are GCH-sets, then the successor aleph  ( aleph `  suc  A ) is equinumerous to the powerset of  ( aleph `  A
). (Contributed by Mario Carneiro, 31-May-2015.)
 |-  ( ( A  e.  On  /\  ( aleph `  A )  e. GCH  /\  ( aleph ` 
 suc  A )  e. GCH )  ->  ( aleph `  suc  A ) 
 ~~  ~P ( aleph `  A ) )
 
Theoremhargch 8179 If  A  +  ~~  ~P A, then  A is a GCH-set. The much simpler converse to gchhar 8173. (Contributed by Mario Carneiro, 2-Jun-2015.)
 |-  ( (har `  A )  ~~  ~P A  ->  A  e. GCH )
 
Theoremalephgch 8180 If  ( aleph `  suc  A ) is equinumerous to the powerset of  ( aleph `  A
), then  ( aleph `  A
) is a GCH-set. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ( aleph `  suc  A )  ~~  ~P ( aleph `  A )  ->  ( aleph `  A )  e. GCH )
 
Theoremgch2 8181 It is sufficient to require that all alephs are GCH-sets to ensure the full generalized continuum hypothesis. (The proof uses the Axiom of Regularity.) (Contributed by Mario Carneiro, 15-May-2015.)
 |-  (GCH  =  _V  <->  ran  aleph  C_ GCH )
 
Theoremgch3 8182 An equivalent formulation of the generalized continuum hypothesis. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  (GCH  =  _V  <->  A. x  e.  On  ( aleph `  suc  x ) 
 ~~  ~P ( aleph `  x ) )
 
Theoremgch-kn 8183* The equivalence of two versions of the Generalized Continuum Hypothesis. The right-hand side is the standard version in the literature. The left-hand side is a version devised by Kannan Nambiar, which he calls the Axiom of Combinatorial Sets. For the notation and motivation behind this axiom, see his paper, "Derivation of Continuum Hypothesis from Axiom of Combinatorial Sets," available at http://www.e-atheneum.net/science/derivation_ch.pdf. The equivalence of the two sides provides a negative answer to Open Problem 2 in http://www.e-atheneum.net/science/open_problem_print.pdf. The key idea in the proof below is to equate both sides of alephexp2 8083 to the successor aleph using enen2 6887. (Contributed by NM, 1-Oct-2004.)
 |-  ( A  e.  On  ->  ( ( aleph `  suc  A )  ~~  { x  |  ( x  C_  ( aleph `  A )  /\  x  ~~  ( aleph `  A )
 ) }  <->  ( aleph `  suc  A )  ~~  ( 2o 
 ^m  ( aleph `  A ) ) ) )
 
PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY

Here we introduce Tarski-Grothendieck (TG) set theory, named after mathematicians Alfred Tarski and Alexander Grothendieck. TG theory extends ZFC with the TG Axiom ax-groth 8325, which states that for every set  x there is an inaccessible cardinal  y such that  y is not in  x. The addition of this axiom to ZFC set theory provides a framework for category theory, thus for all practical purposes giving us a complete foundation for "all of mathematics."

We first introduce the concept of inaccessibles, including Weakly and strongly inaccessible cardinals (df-wina 8186 and df-ina 8187 respectively), Tarski's classes (df-tsk 8251), and a Grothendieck's universe (df-gru 8293). We then introduce the Tarski's axiom ax-groth 8325 and prove various properties from that.

 
4.1  Inaccessibles
 
4.1.1  Weakly and strongly inaccessible cardinals
 
Syntaxcwina 8184 The class of weak inaccessibles.
 class  Inacc W
 
Syntaxcina 8185 The class of strong inaccessibles.
 class  Inacc
 
Definitiondf-wina 8186* An ordinal is weakly inaccessible iff it is a regular limit cardinal. Note that our definition allows  om as a weakly inacessible cardinal. (Contributed by Mario Carneiro, 22-Jun-2013.)
 |- 
 Inacc W  =  { x  |  ( x  =/= 
 (/)  /\  ( cf `  x )  =  x  /\  A. y  e.  x  E. z  e.  x  y  ~<  z ) }
 
Definitiondf-ina 8187* An ordinal is strongly inaccessible iff it is a regular strong limit cardinal, which is to say that it dominates the powersets of every smaller ordinal. (Contributed by Mario Carneiro, 22-Jun-2013.)
 |- 
 Inacc  =  { x  |  ( x  =/=  (/)  /\  ( cf `  x )  =  x  /\  A. y  e.  x  ~P y  ~<  x ) }
 
Theoremelwina 8188* Conditions of weak inaccessibility. (Contributed by Mario Carneiro, 22-Jun-2013.)
 |-  ( A  e.  Inacc W  <-> 
 ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  E. y  e.  A  x  ~<  y ) )
 
Theoremelina 8189* Conditions of strong inaccessibility. (Contributed by Mario Carneiro, 22-Jun-2013.)
 |-  ( A  e.  Inacc  <->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  ~P x  ~<  A ) )
 
Theoremwinaon 8190 A weakly inaccessible cardinal is an ordinal. (Contributed by Mario Carneiro, 29-May-2014.)
 |-  ( A  e.  Inacc W 
 ->  A  e.  On )
 
Theoreminawinalem 8191* Lemma for inawina 8192. (Contributed by Mario Carneiro, 8-Jun-2014.)
 |-  ( A  e.  On  ->  ( A. x  e.  A  ~P x  ~<  A 
 ->  A. x  e.  A  E. y  e.  A  x  ~<  y ) )
 
Theoreminawina 8192 Every strongly inaccessible cardinal is weakly inaccessible. (Contributed by Mario Carneiro, 29-May-2014.)
 |-  ( A  e.  Inacc  ->  A  e.  Inacc W )
 
Theoremomina 8193  om is a strongly inaccessible cardinal. (Many definitions of "inaccessible" explicitly disallow  om as an inaccessible cardinal, but this choice allows us to reuse our results for inaccessibles for  om.) (Contributed by Mario Carneiro, 29-May-2014.)
 |- 
 om  e.  Inacc
 
Theoremwinacard 8194 A weakly inaccessible cardinal is a cardinal. (Contributed by Mario Carneiro, 29-May-2014.)
 |-  ( A  e.  Inacc W 
 ->  ( card `  A )  =  A )
 
Theoremwinainflem 8195* A weakly inaccessible cardinal is infinite. (Contributed by Mario Carneiro, 29-May-2014.)
 |-  ( ( A  =/=  (/)  /\  A  e.  On  /\  A. x  e.  A  E. y  e.  A  x  ~<  y )  ->  om  C_  A )
 
Theoremwinainf 8196 A weakly inaccessible cardinal is infinite. (Contributed by Mario Carneiro, 29-May-2014.)
 |-  ( A  e.  Inacc W 
 ->  om  C_  A )
 
Theoremwinalim 8197 A weakly inaccessible cardinal is a limit ordinal. (Contributed by Mario Carneiro, 29-May-2014.)
 |-  ( A  e.  Inacc W 
 ->  Lim  A )
 
Theoremwinalim2 8198* A nontrivial weakly inaccessible cardinal is a limit aleph. (Contributed by Mario Carneiro, 29-May-2014.)
 |-  ( ( A  e.  Inacc W  /\  A  =/=  om )  ->  E. x ( (
 aleph `  x )  =  A  /\  Lim  x ) )
 
Theoremwinafp 8199 A nontrivial weakly inaccessible cardinal is a fixed point of the aleph function. (Contributed by Mario Carneiro, 29-May-2014.)
 |-  ( ( A  e.  Inacc W  /\  A  =/=  om )  ->  ( aleph `  A )  =  A )
 
Theoremwinafpi 8200 This theorem, which states that a nontrivial inaccessible cardinal is its own aleph number, is stated here in inference form, where the assumptions are in the hypotheses rather than an antecedent. Often, we use dedth 3511 to turn this type of statement into the closed form statement winafp 8199, but in this case, since it is consistent with ZFC that there are no nontrivial inaccessible cardinals, it is not possible to prove winafp 8199 using this theorem and dedth 3511, in ZFC. (You can prove this if you use ax-groth 8325, though.) (Contributed by Mario Carneiro, 28-May-2014.)
 |-  A  e.  Inacc W   &    |-  A  =/=  om   =>    |-  ( aleph `  A )  =  A
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