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Theorem List for Metamath Proof Explorer - 8201-8300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremalephadd 8201 The sum of two alephs is their maximum. Equation 6.1 of [Jech] p. 42. (Contributed by NM, 29-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( aleph `  A )  +c  ( aleph `  B ) )  ~~  ( (
 aleph `  A )  u.  ( aleph `  B )
 )
 
Theoremalephmul 8202 The product of two alephs is their maximum. Equation 6.1 of [Jech] p. 42. (Contributed by NM, 29-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( aleph `  A )  X.  ( aleph `  B ) )  ~~  ( ( aleph `  A )  u.  ( aleph `  B ) ) )
 
Theoremalephexp1 8203 An exponentiation law for alephs. Lemma 6.1 of [Jech] p. 42. (Contributed by NM, 29-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B )  ->  (
 ( aleph `  A )  ^m  ( aleph `  B )
 )  ~~  ( 2o  ^m  ( aleph `  B )
 ) )
 
Theoremalephsuc3 8204* An alternate representation of a successor aleph. Compare alephsuc 7697 and alephsuc2 7709. Equality can be obtained by taking the  card of the right-hand side then using alephcard 7699 and carden 8175. (Contributed by NM, 23-Oct-2004.)
 |-  ( A  e.  On  ->  ( aleph `  suc  A ) 
 ~~  { x  e.  On  |  x  ~~  ( aleph `  A ) } )
 
Theoremalephexp2 8205* An expression equinumerous to 2 to an aleph power. The proof equates the two laws for cardinal exponentiation alephexp1 8203 (which works if the base is less than or equal to the exponent) and infmap 8200 (which works if the exponent is less than or equal to the base). They can be equated only when the base is equal to the exponent, and this is the result. (Contributed by NM, 23-Oct-2004.)
 |-  ( A  e.  On  ->  ( 2o  ^m  ( aleph `  A ) )  ~~  { x  |  ( x 
 C_  ( aleph `  A )  /\  x  ~~  ( aleph `  A ) ) }
 )
 
3.2.5  Cofinality using Axiom of Choice
 
Theoremalephreg 8206 A successor aleph is regular. Theorem 11.15 of [TakeutiZaring] p. 103. (Contributed by Mario Carneiro, 9-Mar-2013.)
 |-  ( cf `  ( aleph `  suc  A ) )  =  ( aleph `  suc  A )
 
Theorempwcfsdom 8207* A corollary of Konig's Theorem konigth 8193. Theorem 11.28 of [TakeutiZaring] p. 108. (Contributed by Mario Carneiro, 20-Mar-2013.)
 |-  H  =  ( y  e.  ( cf `  ( aleph `  A ) )  |->  (har `  ( f `  y
 ) ) )   =>    |-  ( aleph `  A )  ~<  ( ( aleph `  A )  ^m  ( cf `  ( aleph `  A ) ) )
 
Theoremcfpwsdom 8208 A corollary of Konig's Theorem konigth 8193. Theorem 11.29 of [TakeutiZaring] p. 108. (Contributed by Mario Carneiro, 20-Mar-2013.)
 |-  B  e.  _V   =>    |-  ( 2o  ~<_  B  ->  (
 aleph `  A )  ~<  (
 cf `  ( card `  ( B  ^m  ( aleph `  A ) ) ) ) )
 
Theoremalephom 8209 From canth2 7016, we know that  (
aleph `  0 )  < 
( 2 ^ om ), but we cannot prove that  ( 2 ^ om )  =  ( aleph `  1 ) (this is the Continuum Hypothesis), nor can we prove that it is less than any bound whatsoever (i.e. the statement  ( aleph `  A )  <  ( 2 ^ om ) is consistent for any ordinal  A). However, we can prove that  ( 2 ^ om ) is not equal to  ( aleph `  om ), nor  ( aleph `  ( aleph `  om ) ), on cofinality grounds, because by Konig's Theorem konigth 8193 (in the form of cfpwsdom 8208), 
( 2 ^ om ) has uncountable cofinality, which eliminates limit alephs like 
( aleph `  om ). (The first limit aleph that is not eliminated is  (
aleph `  ( aleph `  1
) ), which has cofinality  ( aleph `  1 ).) (Contributed by Mario Carneiro, 21-Mar-2013.)
 |-  ( card `  ( 2o  ^m 
 om ) )  =/=  ( aleph `  om )
 
Theoremsmobeth 8210 The beth function is strictly monotone. This function is not strictly the beth function, but rather bethA is the same as  ( card `  ( R1 `  ( om  +o  A ) ) ), since conventionally we start counting at the first infinite level, and ignore the finite levels. (Contributed by Mario Carneiro, 6-Jun-2013.) (Revised by Mario Carneiro, 2-Jun-2015.)
 |- 
 Smo  ( card  o.  R1 )
 
3.3  ZFC Axioms with no distinct variable requirements
 
Theoremnd1 8211 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 1-Jan-2002.)
 |-  ( A. x  x  =  y  ->  -.  A. x  y  e.  z
 )
 
Theoremnd2 8212 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 1-Jan-2002.)
 |-  ( A. x  x  =  y  ->  -.  A. x  z  e.  y
 )
 
Theoremnd3 8213 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.)
 |-  ( A. x  x  =  y  ->  -.  A. z  x  e.  y
 )
 
Theoremnd4 8214 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.)
 |-  ( A. x  x  =  y  ->  -.  A. z  y  e.  x )
 
Theoremaxextnd 8215 A version of the Axiom of Extensionality with no distinct variable conditions. (Contributed by NM, 14-Aug-2003.)
 |- 
 E. x ( ( x  e.  y  <->  x  e.  z
 )  ->  y  =  z )
 
Theoremaxrepndlem1 8216* Lemma for the Axiom of Replacement with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)
 |-  ( -.  A. y  y  =  z  ->  E. x ( E. y A. z ( ph  ->  z  =  y )  ->  A. z ( z  e.  x  <->  E. x ( x  e.  y  /\  A. y ph ) ) ) )
 
Theoremaxrepndlem2 8217 Lemma for the Axiom of Replacement with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
 |-  ( ( ( -. 
 A. x  x  =  y  /\  -.  A. x  x  =  z
 )  /\  -.  A. y  y  =  z )  ->  E. x ( E. y A. z ( ph  ->  z  =  y ) 
 ->  A. z ( z  e.  x  <->  E. x ( x  e.  y  /\  A. y ph ) ) ) )
 
Theoremaxrepnd 8218 A version of the Axiom of Replacement with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)
 |- 
 E. x ( E. y A. z ( ph  ->  z  =  y ) 
 ->  A. z ( A. y  z  e.  x  <->  E. x ( A. z  x  e.  y  /\  A. y ph ) ) )
 
Theoremaxunndlem1 8219* Lemma for the Axiom of Union with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)
 |- 
 E. x A. y
 ( E. x ( y  e.  x  /\  x  e.  z )  ->  y  e.  x )
 
Theoremaxunnd 8220 A version of the Axiom of Union with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)
 |- 
 E. x A. y
 ( E. x ( y  e.  x  /\  x  e.  z )  ->  y  e.  x )
 
Theoremaxpowndlem1 8221 Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.)
 |-  ( A. x  x  =  y  ->  ( -.  x  =  y  ->  E. x A. y
 ( A. x ( E. z  x  e.  y  ->  A. y  x  e.  z )  ->  y  e.  x ) ) )
 
Theoremaxpowndlem2 8222* Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
 |-  ( -.  A. x  x  =  y  ->  ( -.  A. x  x  =  z  ->  ( -.  x  =  y  ->  E. x A. y
 ( A. x ( E. z  x  e.  y  ->  A. y  x  e.  z )  ->  y  e.  x ) ) ) )
 
Theoremaxpowndlem3 8223* 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 8224 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 8225 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 8226 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 8227* 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 8228 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 8229* Lemma for the Axiom of Infinity with no distinct variable conditions. (New usage is discouraged.) (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 8230 A version of the Axiom of Infinity with no distinct variable conditions. (New usage is discouraged.) (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 8231 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 8232 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 8233 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 8234* Lemma for the Axiom of Choice with no distinct variable conditions. (New usage is discouraged.) (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 8235* 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 8236 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 8237* Axiom of Extensionality ax-ext 2266, 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 8238* Axiom of Replacement ax-rep 4133, 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 8239* Axiom of Union ax-un 4514, 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 8240* Axiom of Power Sets ax-pow 4190, reproved from conditionless ZFC axioms. The proof uses the "Axiom of Twoness," dtru 4203. (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 8241* Axiom of Regularity ax-reg 7308, 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 8242* Axiom of Infinity ax-inf 7341, reproved from conditionless ZFC axioms. Since we have already reproved Extensionality, Replacement, and Power Sets above, we are justified in referencing theorem el 4194 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 8243* Axiom of Choice ax-ac 8087, 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 8244 Extend class notation to include the collection of sets that satisfy the GCH.
 class GCH
 
Definitiondf-gch 8245* 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 8246* 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 8247 A finite set is a GCH-set. (Contributed by Mario Carneiro, 15-May-2015.)
 |- 
 Fin  C_ GCH
 
Theoremgchi 8248 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 8249 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 8250 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 8251 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 8252 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 8253 Under certain conditions, a GCH-set can demonstrate trichotomy of dominance. Lemma for gchac 8297. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ( A  e. GCH  /\  ( A  +c  A )  ~~  A  /\  B  ~<_  ~P A )  ->  ( A 
 ~<_  B  \/  B  ~<_  A ) )
 
Theoremfpwwe2cbv 8254* Lemma for fpwwe2 8267. (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 8255* Lemma for fpwwe2 8267. (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 8256* Lemma for fpwwe2 8267. (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 8257* Lemma for fpwwe2 8267. (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 8258* Lemma for fpwwe2 8267. (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 8259* Lemma for fpwwe2 8267. (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 8260* Lemma for fpwwe2 8267. (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 8261* Lemma for fpwwe2 8267. 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 8262* Lemma for fpwwe2 8267. 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 8263* Lemma for fpwwe2 8267. 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 8264* Lemma for fpwwe2 8267. (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 8265* Lemma for fpwwe2 8267. (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 8266* Lemma for fpwwe2 8267. (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 8267* 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 7659. 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 8268* Lemma for fpwwe 8270. (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 8269* Lemma for fpwwe 8270. (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 8270* Given any function  F from the powerset of  A to  A, canth2 7016 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 7659. 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 8271* An "effective" form of Cantor's theorem canth 6296. 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 8272* Lemma for canthnum 8273. (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 8273 The set of well-orderable subsets of a set  A strictly dominates  A. A stronger form of canth2 7016. 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 8274* Lemma for canthnum 8273. (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 8275* The set of well-orders of a set  A strictly dominates  A. A stronger form of canth2 7016. 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 8276 Lemma for canthp1 8278. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( 1o  ~<  A  ->  ( A  +c  2o )  ~<_  ~P A )
 
Theoremcanthp1lem2 8277* Lemma for canthp1 8278. (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 8278 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 8279 The exclusion of finite sets from consideration in df-gch 8245 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 8280 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 8281 An infinite GCH-set is Dedekind-infinite. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  ( ( A  e. GCH  /\ 
 -.  A  e.  Fin )  ->  om  ~<_  A )
 
Theorempwfseqlem1 8282* Lemma for pwfseq 8288. 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 8283* Lemma for pwfseq 8288. (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 8284* Lemma for pwfseq 8288. Using the construction  D from pwfseqlem1 8282, 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 8285* Lemma for pwfseqlem4 8286. (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 8286* Lemma for pwfseq 8288. Derive a final contradiction from the function  F in pwfseqlem3 8284. Applying fpwwe2 8267 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 8287* Lemma for pwfseq 8288. Although in some ways pwfseqlem4 8286 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 7656. 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 7644. 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 7411), which can be used to construct a pairing function explicitly using properties of the ordinal exponential (infxpenc 7647). (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 8288* 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 8289 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 8290 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 8291 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 8292 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 8293 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 8294 Lemma for gchac 8297 (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 8295 A "local" form of gchac 8297. 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 8296 A "local" form of gchac 8297. 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 8297 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 8298 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 8299 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 8300 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 ) )
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