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Theorem List for Metamath Proof Explorer - 8201-8300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremaxacndlem5 8201* Lemma for the Axiom of Choice with no distinct variable conditions. (New usage is discouraged.) (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 8202 A version of the Axiom of Choice with no distinct variable conditions. (New usage is discouraged.) (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 8203* Axiom of Extensionality ax-ext 2239, 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 8204* Axiom of Replacement ax-rep 4105, 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 8205* Axiom of Union ax-un 4484, 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 8206* Axiom of Power Sets ax-pow 4160, reproved from conditionless ZFC axioms. The proof uses the "Axiom of Twoness," dtru 4173. (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 8207* Axiom of Regularity ax-reg 7274, 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 8208* Axiom of Infinity ax-inf 7307, reproved from conditionless ZFC axioms. Since we have already reproved Extensionality, Replacement, and Power Sets above, we are justified in referencing theorem el 4164 in the proof. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by NM, 15-Aug-2003.)
 |- 
 E. y ( x  e.  y  /\  A. z ( z  e.  y  ->  E. w ( z  e.  w  /\  w  e.  y
 ) ) )
 
Theoremzfcndac 8209* Axiom of Choice ax-ac 8053, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (New usage is discouraged.) (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 8210 Extend class notation to include the collection of sets that satisfy the GCH.
 class GCH
 
Definitiondf-gch 8211* 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 8212* 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 8213 A finite set is a GCH-set. (Contributed by Mario Carneiro, 15-May-2015.)
 |- 
 Fin  C_ GCH
 
Theoremgchi 8214 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 8215 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 8216 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 8217 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 8218 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 8219 Under certain conditions, a GCH-set can demonstrate trichotomy of dominance. Lemma for gchac 8263. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ( A  e. GCH  /\  ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A )  ->  ( A 
 ~<_  B  \/  B  ~<_  A ) )
 
Theoremfpwwe2cbv 8220* Lemma for fpwwe2 8233. (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 8221* Lemma for fpwwe2 8233. (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 8222* Lemma for fpwwe2 8233. (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 8223* Lemma for fpwwe2 8233. (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 8224* Lemma for fpwwe2 8233. (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 8225* Lemma for fpwwe2 8233. (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 8226* Lemma for fpwwe2 8233. (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 8227* Lemma for fpwwe2 8233. 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 8228* Lemma for fpwwe2 8233. 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 8229* Lemma for fpwwe2 8233. 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 8230* Lemma for fpwwe2 8233. (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 8231* Lemma for fpwwe2 8233. (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 8232* Lemma for fpwwe2 8233. (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 8233* 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 7625. 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 8234* Lemma for fpwwe 8236. (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 8235* Lemma for fpwwe 8236. (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 8236* Given any function  F from the powerset of  A to  A, canth2 6982 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 7625. 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 8237* An "effective" form of Cantor's theorem canth 6260. 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 8238* Lemma for canthnum 8239. (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 8239 The set of well-orderable subsets of a set  A strictly dominates  A. A stronger form of canth2 6982. 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 8240* Lemma for canthnum 8239. (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 8241* The set of well-orders of a set  A strictly dominates  A. A stronger form of canth2 6982. 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 8242 Lemma for canthp1 8244. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( 1o  ~<  A  ->  ( A  +c  2o )  ~<_  ~P A )
 
Theoremcanthp1lem2 8243* Lemma for canthp1 8244. (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 8244 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 8245 The exclusion of finite sets from consideration in df-gch 8211 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 8246 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 8247 An infinite GCH-set is Dedekind-infinite. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  ( ( A  e. GCH  /\ 
 -.  A  e.  Fin )  ->  om  ~<_  A )
 
Theorempwfseqlem1 8248* Lemma for pwfseq 8254. 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 8249* Lemma for pwfseq 8254. (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 8250* Lemma for pwfseq 8254. Using the construction  D from pwfseqlem1 8248, 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 8251* Lemma for pwfseqlem4 8252. (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 8252* Lemma for pwfseq 8254. Derive a final contradiction from the function  F in pwfseqlem3 8250. Applying fpwwe2 8233 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 8253* Lemma for pwfseq 8254. Although in some ways pwfseqlem4 8252 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 7622. 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 7610. 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 7377), which can be used to construct a pairing function explicitly using properties of the ordinal exponential (infxpenc 7613). (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 8254* 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 8255 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 8256 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 8257 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 8258 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 8259 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 8260 Lemma for gchac 8263 (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 8261 A "local" form of gchac 8263. 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 8262 A "local" form of gchac 8263. 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 8263 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 8264 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 8265 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 8266 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 8267 If  A  +  ~~  ~P A, then  A is a GCH-set. The much simpler converse to gchhar 8261. (Contributed by Mario Carneiro, 2-Jun-2015.)
 |-  ( (har `  A )  ~~  ~P A  ->  A  e. GCH )
 
Theoremalephgch 8268 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 8269 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 8270 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 8271* 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 8171 to the successor aleph using enen2 6970. (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 8413, 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 8274 and df-ina 8275 respectively), Tarski's classes (df-tsk 8339), and a Grothendieck's universe (df-gru 8381). We then introduce the Tarski's axiom ax-groth 8413 and prove various properties from that.

 
4.1  Inaccessibles
 
4.1.1  Weakly and strongly inaccessible cardinals
 
Syntaxcwina 8272 The class of weak inaccessibles.
 class  Inacc W
 
Syntaxcina 8273 The class of strong inaccessibles.
 class  Inacc
 
Definitiondf-wina 8274* 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 8275* 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 8276* 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 8277* 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 8278 A weakly inaccessible cardinal is an ordinal. (Contributed by Mario Carneiro, 29-May-2014.)
 |-  ( A  e.  Inacc W 
 ->  A  e.  On )
 
Theoreminawinalem 8279* Lemma for inawina 8280. (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 8280 Every strongly inaccessible cardinal is weakly inaccessible. (Contributed by Mario Carneiro, 29-May-2014.)
 |-  ( A  e.  Inacc  ->  A  e.  Inacc W )
 
Theoremomina 8281  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 8282 A weakly inaccessible cardinal is a cardinal. (Contributed by Mario Carneiro, 29-May-2014.)
 |-  ( A  e.  Inacc W 
 ->  ( card `  A )  =  A )
 
Theoremwinainflem 8283* 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 8284 A weakly inaccessible cardinal is infinite. (Contributed by Mario Carneiro, 29-May-2014.)
 |-  ( A  e.  Inacc W 
 ->  om  C_  A )
 
Theoremwinalim 8285 A weakly inaccessible cardinal is a limit ordinal. (Contributed by Mario Carneiro, 29-May-2014.)
 |-  ( A  e.  Inacc W 
 ->  Lim  A )
 
Theoremwinalim2 8286* 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 8287 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 8288 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 3580 to turn this type of statement into the closed form statement winafp 8287, but in this case, since it is consistent with ZFC that there are no nontrivial inaccessible cardinals, it is not possible to prove winafp 8287 using this theorem and dedth 3580, in ZFC. (You can prove this if you use ax-groth 8413, though.) (Contributed by Mario Carneiro, 28-May-2014.)
 |-  A  e.  Inacc W   &    |-  A  =/=  om   =>    |-  ( aleph `  A )  =  A
 
Theoremgchina 8289 Assuming the GCH, weakly and strongly inaccessible cardinals coincide. Theorem 11.20 of [TakeutiZaring] p. 106. (Contributed by Mario Carneiro, 5-Jun-2015.)
 |-  (GCH  =  _V  ->  Inacc W  =  Inacc )
 
4.1.2  Weak universes
 
Syntaxcwun 8290 Extend class definition to include the class of all weak universes.
 class WUni
 
Syntaxcwunm 8291 Extend class definition to include the map whose value is the smallest weak universe.
 class wUniCl
 
Definitiondf-wun 8292* The class of all weak universes. A weak universe is a nonempty transitive class closed under union, pairing, and powerset. The advantage of a weak universe over a Grothendieck universe is that weak universes satisfy the analogue uniwun 8330 of grothtsk 8425 in ZFC. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |- WUni  =  { u  |  ( Tr  u  /\  u  =/= 
 (/)  /\  A. x  e.  u  ( U. x  e.  u  /\  ~P x  e.  u  /\  A. y  e.  u  { x ,  y }  e.  u ) ) }
 
Definitiondf-wunc 8293* A function that maps a set  x to the smallest weak universe that contains the elements of the set. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |- wUniCl  =  ( x  e.  _V  |->  |^|
 { u  e. WUni  |  x  C_  u } )
 
Theoremiswun 8294* Properties of a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( U  e.  V  ->  ( U  e. WUni  <->  ( Tr  U  /\  U  =/=  (/)  /\  A. x  e.  U  ( U. x  e.  U  /\  ~P x  e.  U  /\  A. y  e.  U  { x ,  y }  e.  U ) ) ) )
 
Theoremwuntr 8295 A weak universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( U  e. WUni  ->  Tr  U )
 
Theoremwununi 8296 A weak universe is closed under union. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A  e.  U )   =>    |-  ( ph  ->  U. A  e.  U )
 
Theoremwunpw 8297 A weak universe is closed under powerset. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A  e.  U )   =>    |-  ( ph  ->  ~P A  e.  U )
 
Theoremwunelss 8298 The elements of a weak universe are also subsets of it. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A  e.  U )   =>    |-  ( ph  ->  A 
 C_  U )
 
Theoremwunpr 8299 A weak universe is closed under pairing. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  B  e.  U )   =>    |-  ( ph  ->  { A ,  B }  e.  U )
 
Theoremwunun 8300 A weak universe is closed under binary union. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  B  e.  U )   =>    |-  ( ph  ->  ( A  u.  B )  e.  U )
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