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Theorem List for Metamath Proof Explorer - 8101-8200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremaxdclem2 8101* Lemma for axdc 8102. Using the full Axiom of Choice, we can construct a choice function  g on  ~P dom  x. From this, we can build a sequence  F starting at any value  s  e.  dom  x by repeatedly applying  g to the set  ( F `  x ) (where  x is the value from the previous iteration). (Contributed by Mario Carneiro, 25-Jan-2013.)
 |-  F  =  ( rec ( ( y  e. 
 _V  |->  ( g `  { z  |  y x z } )
 ) ,  s )  |`  om )   =>    |-  ( E. z  s x z  ->  ( ran  x  C_  dom  x  ->  E. f A. n  e. 
 om  ( f `  n ) x ( f `  suc  n ) ) )
 
Theoremaxdc 8102* This theorem derives ax-dc 8026 using ax-ac 8039 and ax-inf 7293. Thus, AC implies DC, but not vice-versa (so that ZFC is strictly stronger than ZF+DC). (Contributed by Mario Carneiro, 25-Jan-2013.)
 |-  ( ( E. y E. z  y x z  /\  ran  x  C_  dom  x )  ->  E. f A. n  e.  om  ( f `  n ) x ( f `  suc  n ) )
 
Theoremfodom 8103 An onto function implies dominance of domain over range. Lemma 10.20 of [Kunen] p. 30. This theorem uses the Axiom of Choice ac7g 8055. AC is not needed for finite sets - see fodomfi 7089. See also fodomnum 7638. (Contributed by NM, 23-Jul-2004.)
 |-  A  e.  _V   =>    |-  ( F : A -onto-> B  ->  B  ~<_  A )
 
Theoremfodomg 8104 An onto function implies dominance of domain over range. (Contributed by NM, 23-Jul-2004.)
 |-  ( A  e.  C  ->  ( F : A -onto-> B  ->  B  ~<_  A ) )
 
Theoremfodomb 8105* Equivalence of an onto mapping and dominance for a non-empty set. Proposition 10.35 of [TakeutiZaring] p. 93. (Contributed by NM, 29-Jul-2004.)
 |-  ( ( A  =/=  (/)  /\  E. f  f : A -onto-> B )  <->  ( (/)  ~<  B  /\  B 
 ~<_  A ) )
 
Theoremwdomac 8106 When assuming AC, weak and usual dominance coincide. It is not known if this is an AC equivalent. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
 |-  ( X  ~<_*  Y  <->  X  ~<_  Y )
 
Theorembrdom3 8107* Equivalence to a dominance relation. (Contributed by NM, 27-Mar-2007.)
 |-  B  e.  _V   =>    |-  ( A  ~<_  B  <->  E. f ( A. x E* y  x f y  /\  A. x  e.  A  E. y  e.  B  y f x ) )
 
Theorembrdom5 8108* An equivalence to a dominance relation. (Contributed by NM, 29-Mar-2007.)
 |-  B  e.  _V   =>    |-  ( A  ~<_  B  <->  E. f ( A. x  e.  B  E* y  x f y  /\  A. x  e.  A  E. y  e.  B  y
 f x ) )
 
Theorembrdom4 8109* An equivalence to a dominance relation. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.)
 |-  B  e.  _V   =>    |-  ( A  ~<_  B  <->  E. f ( A. x  e.  B  E* y  e.  A x f y  /\  A. x  e.  A  E. y  e.  B  y f x ) )
 
Theorembrdom7disj 8110* An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 29-Mar-2007.) (Revised by NM, 16-Jun-2017.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( A  i^i  B )  =  (/)   =>    |-  ( A  ~<_  B  <->  E. f ( A. x  e.  B  E* y  e.  A { x ,  y }  e.  f  /\  A. x  e.  A  E. y  e.  B  { y ,  x }  e.  f
 ) )
 
Theorembrdom6disj 8111* An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 5-Apr-2007.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( A  i^i  B )  =  (/)   =>    |-  ( A  ~<_  B  <->  E. f ( A. x  e.  B  E* y { x ,  y }  e.  f  /\  A. x  e.  A  E. y  e.  B  { y ,  x }  e.  f
 ) )
 
Theoremfin71ac 8112 Once we allow AC, the "strongest" definition of finite set becomes equivalent to the "weakest" and the entire hierarchy collapses. (Contributed by Stefan O'Rear, 29-Oct-2014.)
 |- FinVII  = 
 Fin
 
Theoremimadomg 8113 An image of a function under a set is dominated by the set. Proposition 10.34 of [TakeutiZaring] p. 92. (Contributed by NM, 23-Jul-2004.)
 |-  ( A  e.  B  ->  ( Fun  F  ->  ( F " A )  ~<_  A ) )
 
Theoremfnrndomg 8114 The range of a function is dominated by its domain. (Contributed by NM, 1-Sep-2004.)
 |-  ( A  e.  B  ->  ( F  Fn  A  ->  ran  F  ~<_  A ) )
 
Theoremiunfo 8115* Existence of an onto function from a disjoint union to a union. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 18-Jan-2014.)
 |-  T  =  U_ x  e.  A  ( { x }  X.  B )   =>    |-  ( 2nd  |`  T ) : T -onto-> U_ x  e.  A  B
 
Theoremiundom2g 8116* An upper bound for the cardinality of an disjoint indexed union, with explicit choice principles. 
B depends on  x and should be thought of as  B ( x ). (Contributed by Mario Carneiro, 1-Sep-2015.)
 |-  T  =  U_ x  e.  A  ( { x }  X.  B )   &    |-  ( ph  ->  U_ x  e.  A  ( C  ^m  B )  e. AC  A )   &    |-  ( ph  ->  A. x  e.  A  B  ~<_  C )   =>    |-  ( ph  ->  T  ~<_  ( A  X.  C ) )
 
Theoremiundomg 8117* An upper bound for the cardinality of an indexed union, with explicit choice principles.  B depends on  x and should be thought of as  B ( x ). (Contributed by Mario Carneiro, 1-Sep-2015.)
 |-  T  =  U_ x  e.  A  ( { x }  X.  B )   &    |-  ( ph  ->  U_ x  e.  A  ( C  ^m  B )  e. AC  A )   &    |-  ( ph  ->  A. x  e.  A  B  ~<_  C )   &    |-  ( ph  ->  ( A  X.  C )  e. AC  U_ x  e.  A  B )   =>    |-  ( ph  ->  U_ x  e.  A  B  ~<_  ( A  X.  C ) )
 
Theoremiundom 8118* An upper bound for the cardinality of an indexed union.  C depends on  x and should be thought of as  C ( x ). (Contributed by NM, 26-Mar-2006.)
 |-  ( ( A  e.  V  /\  A. x  e.  A  C  ~<_  B ) 
 ->  U_ x  e.  A  C 
 ~<_  ( A  X.  B ) )
 
Theoremunidom 8119* An upper bound for the cardinality of a union. Theorem 10.47 of [TakeutiZaring] p. 98. (Contributed by NM, 25-Mar-2006.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)
 |-  ( ( A  e.  V  /\  A. x  e.  A  x  ~<_  B ) 
 ->  U. A  ~<_  ( A  X.  B ) )
 
Theoremuniimadom 8120* An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. (Contributed by NM, 25-Mar-2006.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ( Fun  F  /\  A. x  e.  A  ( F `  x )  ~<_  B )  ->  U. ( F " A )  ~<_  ( A  X.  B ) )
 
Theoremuniimadomf 8121* An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 8120 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by NM, 26-Mar-2006.)
 |-  F/_ x F   &    |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ( Fun  F  /\  A. x  e.  A  ( F `  x )  ~<_  B )  ->  U. ( F " A )  ~<_  ( A  X.  B ) )
 
3.2.3  Cardinal number theorems using Axiom of Choice
 
Theoremcardval 8122* The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 7578 for a simpler version of its value. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  A  e.  _V   =>    |-  ( card `  A )  =  |^| { x  e.  On  |  x  ~~  A }
 
Theoremcardid 8123 Any set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  A  e.  _V   =>    |-  ( card `  A )  ~~  A
 
Theoremcardidg 8124 Any set is equinumerous to its cardinal number. Closed theorem form of cardid 8123. (Contributed by David Moews, 1-May-2017.)
 |-  ( A  e.  B  ->  ( card `  A )  ~~  A )
 
Theoremcardidd 8125 Any set is equinumerous to its cardinal number. Deduction form of cardid 8123. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  B )   =>    |-  ( ph  ->  ( card `  A )  ~~  A )
 
Theoremcardf 8126 The cardinality function is a function with domain the well-orderable sets. Assuming AC, this is the universe. (Contributed by Mario Carneiro, 6-Jun-2013.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |- 
 card : _V --> On
 
Theoremcarden 8127 Two sets are equinumerous iff their cardinal numbers are equal. This important theorem expresses the essential concept behind "cardinality" or "size." This theorem appears as Proposition 10.10 of [TakeutiZaring] p. 85, Theorem 7P of [Enderton] p. 197, and Theorem 9 of [Suppes] p. 242 (among others). The Axiom of Choice is required for its proof.

The theory of cardinality can also be developed without AC by introducing "card" as a primitive notion and stating this theorem as an axiom, as is done with the axiom for cardinal numbers in [Suppes] p. 111. Finally, if we allow the Axiom of Regularity, we can avoid AC by defining the cardinal number of a set as the set of all sets equinumerous to it and having least possible rank (see karden 7519). (Contributed by NM, 22-Oct-2003.)

 |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( card `  A )  =  (
 card `  B )  <->  A  ~~  B ) )
 
Theoremcardeq0 8128 Only the empty set has cardinality zero. (Contributed by NM, 23-Apr-2004.)
 |-  ( A  e.  V  ->  ( ( card `  A )  =  (/)  <->  A  =  (/) ) )
 
Theoremunsnen 8129 Equinumerosity of a set with a new element added. (Contributed by NM, 7-Nov-2008.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( -.  B  e.  A  ->  ( A  u.  { B } )  ~~  suc  ( card `  A )
 )
 
Theoremcarddom 8130 Two sets have the dominance relationship iff their cardinalities have the subset relationship. Equation i of [Quine] p. 232. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( card `  A )  C_  ( card `  B )  <->  A  ~<_  B )
 )
 
Theoremcardsdom 8131 Two sets have the strict dominance relationship iff their cardinalities have the membership relationship. Corollary 19.7(2) of [Eisenberg] p. 310. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( card `  A )  e.  ( card `  B )  <->  A  ~<  B ) )
 
Theoremdomtri 8132 Trichotomy law for dominance and strict dominance. This theorem is equivalent to the Axiom of Choice. (Contributed by NM, 4-Jan-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )
 
Theorementric 8133 Trichotomy of equinumerosity and strict dominance. This theorem is equivalent to the Axiom of Choice. Theorem 8 of [Suppes] p. 242. (Contributed by NM, 4-Jan-2004.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  ~<  B  \/  A  ~~  B  \/  B  ~<  A )
 )
 
Theorementri2 8134 Trichotomy of dominance and strict dominance. (Contributed by NM, 4-Jan-2004.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  ~<_  B  \/  B  ~<  A ) )
 
Theorementri3 8135 Trichotomy of dominance. This theorem is equivalent to the Axiom of Choice. Part of Proposition 4.42(d) of [Mendelson] p. 275. (Contributed by NM, 4-Jan-2004.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  ~<_  B  \/  B 
 ~<_  A ) )
 
Theoremsdomsdomcard 8136 A set strictly dominates iff its cardinal strictly dominates. (Contributed by NM, 30-Oct-2003.)
 |-  ( A  ~<  B  <->  A  ~<  ( card `  B ) )
 
Theoremcanth3 8137 Cantor's theorem in terms of cardinals. This theorem tells us that no matter how largei a cardinal number is, there is a still larger cardinal number. Theorem 18.12 of [Monk1] p. 133. (Contributed by NM, 5-Nov-2003.)
 |-  ( A  e.  V  ->  ( card `  A )  e.  ( card `  ~P A ) )
 
Theoreminfxpidm 8138 The cross product of an infinite set with itself is idempotent. This theorem (which is an AC equivalent) provides the basis for infinite cardinal arithmetic. Proposition 10.40 of [TakeutiZaring] p. 95. This proof follows as a corollary of infxpen 7596. (Contributed by NM, 17-Sep-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)
 |-  ( om  ~<_  A  ->  ( A  X.  A ) 
 ~~  A )
 
Theoremondomon 8139* The collection of ordinal numbers dominated by a set is an ordinal number. (In general, not all collections of ordinal numbers are ordinal.) Theorem 56 of [Suppes] p. 227. This theorem can be proved (with a longer proof) without the Axiom of Choice; see hartogs 7213. (Contributed by NM, 7-Nov-2003.) (Proof modification is discouraged.)
 |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<_  A }  e.  On )
 
Theoremcardmin 8140* The smallest ordinal that strictly dominates a set is a cardinal. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 20-Sep-2014.)
 |-  ( A  e.  V  ->  ( card `  |^| { x  e.  On  |  A  ~<  x } )  =  |^| { x  e.  On  |  A  ~<  x } )
 
Theoremficard 8141 A set is finite iff its cardinal is a natural number. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( A  e.  V  ->  ( A  e.  Fin  <->  ( card `  A )  e. 
 om ) )
 
Theoreminfinf 8142 Equivalence between two infiniteness criteria for sets. (Contributed by David Moews, 1-May-2017.)
 |-  ( A  e.  B  ->  ( -.  A  e.  Fin  <->  om  ~<_  A ) )
 
Theoremunirnfdomd 8143 The union of the range of a function from a infinite set into the class of finite sets is dominated by its domain. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  F : T --> Fin )   &    |-  ( ph  ->  -.  T  e.  Fin )   &    |-  ( ph  ->  T  e.  V )   =>    |-  ( ph  ->  U. ran  F  ~<_  T )
 
Theoremkonigthlem 8144* Lemma for konigth 8145. (Contributed by Mario Carneiro, 22-Feb-2013.)
 |-  A  e.  _V   &    |-  S  =  U_ i  e.  A  ( M `  i )   &    |-  P  =  X_ i  e.  A  ( N `  i )   &    |-  D  =  ( i  e.  A  |->  ( a  e.  ( M `
  i )  |->  ( ( f `  a
 ) `  i )
 ) )   &    |-  E  =  ( i  e.  A  |->  ( e `  i ) )   =>    |-  ( A. i  e.  A  ( M `  i )  ~<  ( N `
  i )  ->  S  ~<  P )
 
Theoremkonigth 8145* Konig's Theorem. If  m ( i ) 
~<  n ( i ) for all 
i  e.  A, then  sum_ i  e.  A m ( i )  ~<  prod_ i  e.  A n ( i ), where the sums and products stand in for disjoint union and infinite cartesian product. The version here is proven with regular unions rather than disjoint unions for convenience, but the version with disjoint unions is clearly a special case of this version. The Axiom of Choice is needed for this proof, but it contains AC as a simple corollary (letting  m ( i )  =  (/), this theorem says that an infinite cartesian product of nonempty sets is nonempty), so this is an AC equivalent. Theorem 11.26 of [TakeutiZaring] p. 107. (Contributed by Mario Carneiro, 22-Feb-2013.)
 |-  A  e.  _V   &    |-  S  =  U_ i  e.  A  ( M `  i )   &    |-  P  =  X_ i  e.  A  ( N `  i )   =>    |-  ( A. i  e.  A  ( M `  i )  ~<  ( N `
  i )  ->  S  ~<  P )
 
Theoremalephsucpw 8146 The power set of an aleph dominates the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous, see gch3 8256 or gchaleph2 8252.) (Contributed by NM, 27-Aug-2005.)
 |-  ( aleph `  suc  A )  ~<_  ~P ( aleph `  A )
 
Theoremaleph1 8147 The set exponentiation of 2 to the aleph-zero has cardinality of at least aleph-one. (If we were to assume the Continuum Hypothesis, their cardinalities would be the same.) (Contributed by NM, 7-Jul-2004.)
 |-  ( aleph `  1o )  ~<_  ( 2o  ^m  ( aleph `  (/) ) )
 
Theoremalephval2 8148* An alternate way to express the value of the aleph function for nonzero arguments. Theorem 64 of [Suppes] p. 229. (Contributed by NM, 15-Nov-2003.)
 |-  ( ( A  e.  On  /\  (/)  e.  A ) 
 ->  ( aleph `  A )  =  |^| { x  e. 
 On  |  A. y  e.  A  ( aleph `  y
 )  ~<  x } )
 
Theoremdominfac 8149 A nonempty set that is a subset of its union is infinite. This version is proved from ax-ac 8039. See dominf 8025 for a version proved from ax-cc 8015. (Contributed by NM, 25-Mar-2007.)
 |-  A  e.  _V   =>    |-  ( ( A  =/=  (/)  /\  A  C_  U. A )  ->  om  ~<_  A )
 
3.2.4  Cardinal number arithmetic using Axiom of Choice
 
Theoremiunctb 8150* The countable union of countable sets is countable (indexed union version of unictb 8151). (Contributed by Mario Carneiro, 18-Jan-2014.)
 |-  ( ( A  ~<_  om  /\  A. x  e.  A  B  ~<_  om )  ->  U_ x  e.  A  B  ~<_  om )
 
Theoremunictb 8151* The countable union of countable sets is countable. Theorem 6Q of [Enderton] p. 159. See iunctb 8150 for indexed union version. (Contributed by NM, 26-Mar-2006.)
 |-  ( ( A  ~<_  om  /\  A. x  e.  A  x  ~<_  om )  ->  U. A  ~<_  om )
 
Theoreminfmap 8152* An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. (Contributed by NM, 1-Oct-2004.) (Proof shortened by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( om  ~<_  A  /\  B 
 ~<_  A )  ->  ( A  ^m  B )  ~~  { x  |  ( x 
 C_  A  /\  x  ~~  B ) } )
 
Theoremalephadd 8153 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 8154 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 8155 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 8156* An alternate representation of a successor aleph. Compare alephsuc 7649 and alephsuc2 7661. Equality can be obtained by taking the  card of the right-hand side then using alephcard 7651 and carden 8127. (Contributed by NM, 23-Oct-2004.)
 |-  ( A  e.  On  ->  ( aleph `  suc  A ) 
 ~~  { x  e.  On  |  x  ~~  ( aleph `  A ) } )
 
Theoremalephexp2 8157* An expression equinumerous to 2 to an aleph power. The proof equates the two laws for cardinal exponentiation alephexp1 8155 (which works if the base is less than or equal to the exponent) and infmap 8152 (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 8158 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 8159* A corollary of Konig's Theorem konigth 8145. 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 8160 A corollary of Konig's Theorem konigth 8145. 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 8161 From canth2 6968, 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 8145 (in the form of cfpwsdom 8160), 
( 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 8162 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 8163 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 1-Jan-2002.)
 |-  ( A. x  x  =  y  ->  -.  A. x  y  e.  z
 )
 
Theoremnd2 8164 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 1-Jan-2002.)
 |-  ( A. x  x  =  y  ->  -.  A. x  z  e.  y
 )
 
Theoremnd3 8165 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.)
 |-  ( A. x  x  =  y  ->  -.  A. z  x  e.  y
 )
 
Theoremnd4 8166 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.)
 |-  ( A. x  x  =  y  ->  -.  A. z  y  e.  x )
 
Theoremaxextnd 8167 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 8168* 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 8169 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 8170 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 8171* 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 8172 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 8173 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 8174* 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 8175* 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 8176 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 8177 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 8178 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 8179* 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 8180 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 8181* 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 8182 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 8183 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 8184 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 8185 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 8186* 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 8187* 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 8188 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 8189* Axiom of Extensionality ax-ext 2237, 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 8190* Axiom of Replacement ax-rep 4091, 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 8191* Axiom of Union ax-un 4470, 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 8192* Axiom of Power Sets ax-pow 4146, reproved from conditionless ZFC axioms. The proof uses the "Axiom of Twoness," dtru 4159. (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 8193* Axiom of Regularity ax-reg 7260, 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 8194* Axiom of Infinity ax-inf 7293, reproved from conditionless ZFC axioms. Since we have already reproved Extensionality, Replacement, and Power Sets above, we are justified in referencing theorem el 4150 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 8195* Axiom of Choice ax-ac 8039, 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 8196 Extend class notation to include the collection of sets that satisfy the GCH.
 class GCH
 
Definitiondf-gch 8197* 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 8198* 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 8199 A finite set is a GCH-set. (Contributed by Mario Carneiro, 15-May-2015.)
 |- 
 Fin  C_ GCH
 
Theoremgchi 8200 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 )
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