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Theorem List for Metamath Proof Explorer - 8801-8900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremzornn0 8801* Variant of Zorn's lemma zorn 8800 in which  (/), the union of the empty chain, is not required to be an element of  A. (Contributed by Jeff Madsen, 5-Jan-2011.)
 |-  A  e.  _V   =>    |-  ( ( A  =/=  (/)  /\  A. z ( ( z  C_  A  /\  z  =/=  (/)  /\ [ C.]  Or  z
 )  ->  U. z  e.  A ) )  ->  E. x  e.  A  A. y  e.  A  -.  x  C.  y )
 
Theoremttukeylem1 8802* Lemma for ttukey 8811. Expand out the property of being an element of a property of finite character. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ph  ->  F : ( card `  ( U. A  \  B ) ) -1-1-onto-> ( U. A  \  B ) )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )   =>    |-  ( ph  ->  ( C  e.  A  <->  ( ~P C  i^i  Fin )  C_  A ) )
 
Theoremttukeylem2 8803* Lemma for ttukey 8811. A property of finite character is closed under subsets. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ph  ->  F : ( card `  ( U. A  \  B ) ) -1-1-onto-> ( U. A  \  B ) )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )   =>    |-  ( ( ph  /\  ( C  e.  A  /\  D  C_  C ) ) 
 ->  D  e.  A )
 
Theoremttukeylem3 8804* Lemma for ttukey 8811. (Contributed by Mario Carneiro, 11-May-2015.)
 |-  ( ph  ->  F : ( card `  ( U. A  \  B ) ) -1-1-onto-> ( U. A  \  B ) )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )   &    |-  G  = recs ( (
 z  e.  _V  |->  if ( dom  z  = 
 U. dom  z ,  if ( dom  z  =  (/) ,  B ,  U. ran  z ) ,  (
 ( z `  U. dom  z )  u.  if ( ( ( z `
  U. dom  z )  u.  { ( F `
  U. dom  z ) } )  e.  A ,  { ( F `  U.
 dom  z ) } ,  (/) ) ) ) ) )   =>    |-  ( ( ph  /\  C  e.  On )  ->  ( G `  C )  =  if ( C  =  U. C ,  if ( C  =  (/) ,  B ,  U. ( G " C ) ) ,  ( ( G `  U. C )  u.  if ( ( ( G `
  U. C )  u. 
 { ( F `  U. C ) } )  e.  A ,  { ( F `  U. C ) } ,  (/) ) ) ) )
 
Theoremttukeylem4 8805* Lemma for ttukey 8811. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ph  ->  F : ( card `  ( U. A  \  B ) ) -1-1-onto-> ( U. A  \  B ) )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )   &    |-  G  = recs ( (
 z  e.  _V  |->  if ( dom  z  = 
 U. dom  z ,  if ( dom  z  =  (/) ,  B ,  U. ran  z ) ,  (
 ( z `  U. dom  z )  u.  if ( ( ( z `
  U. dom  z )  u.  { ( F `
  U. dom  z ) } )  e.  A ,  { ( F `  U.
 dom  z ) } ,  (/) ) ) ) ) )   =>    |-  ( ph  ->  ( G `  (/) )  =  B )
 
Theoremttukeylem5 8806* Lemma for ttukey 8811. The  G function forms a (transfinitely long) chain of inclusions. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ph  ->  F : ( card `  ( U. A  \  B ) ) -1-1-onto-> ( U. A  \  B ) )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )   &    |-  G  = recs ( (
 z  e.  _V  |->  if ( dom  z  = 
 U. dom  z ,  if ( dom  z  =  (/) ,  B ,  U. ran  z ) ,  (
 ( z `  U. dom  z )  u.  if ( ( ( z `
  U. dom  z )  u.  { ( F `
  U. dom  z ) } )  e.  A ,  { ( F `  U.
 dom  z ) } ,  (/) ) ) ) ) )   =>    |-  ( ( ph  /\  ( C  e.  On  /\  D  e.  On  /\  C  C_  D ) )  ->  ( G `  C ) 
 C_  ( G `  D ) )
 
Theoremttukeylem6 8807* Lemma for ttukey 8811. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ph  ->  F : ( card `  ( U. A  \  B ) ) -1-1-onto-> ( U. A  \  B ) )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )   &    |-  G  = recs ( (
 z  e.  _V  |->  if ( dom  z  = 
 U. dom  z ,  if ( dom  z  =  (/) ,  B ,  U. ran  z ) ,  (
 ( z `  U. dom  z )  u.  if ( ( ( z `
  U. dom  z )  u.  { ( F `
  U. dom  z ) } )  e.  A ,  { ( F `  U.
 dom  z ) } ,  (/) ) ) ) ) )   =>    |-  ( ( ph  /\  C  e.  suc  ( card `  ( U. A  \  B ) ) )  ->  ( G `  C )  e.  A )
 
Theoremttukeylem7 8808* Lemma for ttukey 8811. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ph  ->  F : ( card `  ( U. A  \  B ) ) -1-1-onto-> ( U. A  \  B ) )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )   &    |-  G  = recs ( (
 z  e.  _V  |->  if ( dom  z  = 
 U. dom  z ,  if ( dom  z  =  (/) ,  B ,  U. ran  z ) ,  (
 ( z `  U. dom  z )  u.  if ( ( ( z `
  U. dom  z )  u.  { ( F `
  U. dom  z ) } )  e.  A ,  { ( F `  U.
 dom  z ) } ,  (/) ) ) ) ) )   =>    |-  ( ph  ->  E. x  e.  A  ( B  C_  x  /\  A. y  e.  A  -.  x  C.  y ) )
 
Theoremttukey2g 8809* The Teichmüller-Tukey Lemma ttukey 8811 with a slightly stronger conclusion: we can set up the maximal element of  A so that it also contains some given  B  e.  A as a subset. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ( U. A  e.  dom  card  /\  B  e.  A  /\  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )  ->  E. x  e.  A  ( B  C_  x  /\  A. y  e.  A  -.  x  C.  y ) )
 
Theoremttukeyg 8810* The Teichmüller-Tukey Lemma ttukey 8811 stated with the "choice" as an antecedent (the hypothesis  U. A  e.  dom  card says that  U. A is well-orderable). (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ( U. A  e.  dom  card  /\  A  =/=  (/)  /\  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )  ->  E. x  e.  A  A. y  e.  A  -.  x  C.  y )
 
Theoremttukey 8811* The Teichmüller-Tukey Lemma, an Axiom of Choice equivalent. If  A is a nonempty collection of finite character, then  A has a maximal element with respect to inclusion. Here "finite character" means that  x  e.  A iff every finite subset of  x is in  A. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  A  e.  _V   =>    |-  ( ( A  =/=  (/)  /\  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )  ->  E. x  e.  A  A. y  e.  A  -.  x  C.  y )
 
Theoremaxdclem 8812* Lemma for axdc 8814. (Contributed by Mario Carneiro, 25-Jan-2013.)
 |-  F  =  ( rec ( ( y  e. 
 _V  |->  ( g `  { z  |  y x z } )
 ) ,  s )  |`  om )   =>    |-  ( ( A. y  e.  ~P  dom  x (
 y  =/=  (/)  ->  (
 g `  y )  e.  y )  /\  ran  x 
 C_  dom  x  /\  E. z ( F `  K ) x z )  ->  ( K  e.  om  ->  ( F `  K ) x ( F `  suc  K ) ) )
 
Theoremaxdclem2 8813* Lemma for axdc 8814. 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 8814* This theorem derives ax-dc 8739 using ax-ac 8752 and ax-inf 7969. Thus, AC implies DC, but not vice-versa (so that ZFC is strictly stronger than ZF+DC). (New usage is discouraged.) (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 8815 An onto function implies dominance of domain over range. Lemma 10.20 of [Kunen] p. 30. This theorem uses the Axiom of Choice ac7g 8767. AC is not needed for finite sets - see fodomfi 7714. See also fodomnum 8351. (Contributed by NM, 23-Jul-2004.)
 |-  A  e.  _V   =>    |-  ( F : A -onto-> B  ->  B  ~<_  A )
 
Theoremfodomg 8816 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 8817* Equivalence of an onto mapping and dominance for a nonempty 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 8818 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 8819* 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 8820* 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 8821* 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 8822* 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 8823* 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 8824 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 8825 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 8826 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 8827* 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 8828* An upper bound for the cardinality of a 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 8829* 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 8830* 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 8831* 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 8832* 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 8833* An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 8832 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 8834* The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 8285 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 8835 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 8836 Any set is equinumerous to its cardinal number. Closed theorem form of cardid 8835. (Contributed by David Moews, 1-May-2017.)
 |-  ( A  e.  B  ->  ( card `  A )  ~~  A )
 
Theoremcardidd 8837 Any set is equinumerous to its cardinal number. Deduction form of cardid 8835. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  B )   =>    |-  ( ph  ->  ( card `  A )  ~~  A )
 
Theoremcardf 8838 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 8839 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. Related theorems are hasheni 12323 and the finite-set-only hashen 12322.

This theorem is also known as Hume's Principle. Gottlob Frege's two-volume Grundgesetze der Arithmetik used his Basic Law V to prove this theorem. Unfortunately Basic Law V caused Frege's system to be inconsistent because it was subject to Russell's paradox (see ru 3251). Later scholars have found that Frege primarily used Basic Law V to Hume's Principle. If Basic Law V is replaced by Hume's Principle in Frege's system, much of Frege's work is restored. Grundgesetze der Arithmetik, once Basic Law V is replaced, proves "Frege's theorem" (the Peano axioms of arithmetic can be derived in second-order logic from Hume's principle). See https://plato.stanford.edu/entries/frege-theorem . We take a different approach, using first-order logic and ZFC, to prove the Peano axioms of arithmetic.

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 the least possible rank (see karden 8226). (Contributed by NM, 22-Oct-2003.)

 |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( card `  A )  =  (
 card `  B )  <->  A  ~~  B ) )
 
Theoremcardeq0 8840 Only the empty set has cardinality zero. (Contributed by NM, 23-Apr-2004.)
 |-  ( A  e.  V  ->  ( ( card `  A )  =  (/)  <->  A  =  (/) ) )
 
Theoremunsnen 8841 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 8842 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 8843 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 8844 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 8845 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 8846 Trichotomy of dominance and strict dominance. (Contributed by NM, 4-Jan-2004.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  ~<_  B  \/  B  ~<  A ) )
 
Theorementri3 8847 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 8848 A set strictly dominates iff its cardinal strictly dominates. (Contributed by NM, 30-Oct-2003.)
 |-  ( A  ~<  B  <->  A  ~<  ( card `  B ) )
 
Theoremcanth3 8849 Cantor's theorem in terms of cardinals. This theorem tells us that no matter how large 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 8850 The Cartesian 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 8305. (Contributed by NM, 17-Sep-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)
 |-  ( om  ~<_  A  ->  ( A  X.  A ) 
 ~~  A )
 
Theoremondomon 8851* 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 7884. (Contributed by NM, 7-Nov-2003.) (Proof modification is discouraged.)
 |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<_  A }  e.  On )
 
Theoremcardmin 8852* 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 8853 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 8854 Equivalence between two infiniteness criteria for sets. (Contributed by David Moews, 1-May-2017.)
 |-  ( A  e.  B  ->  ( -.  A  e.  Fin  <->  om  ~<_  A ) )
 
Theoremunirnfdomd 8855 The union of the range of a function from an 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 8856* Lemma for konigth 8857. (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 8857* 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 8858 The power set of an aleph dominates the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous, see gch3 8965 or gchaleph2 8961.) (Contributed by NM, 27-Aug-2005.)
 |-  ( aleph `  suc  A )  ~<_  ~P ( aleph `  A )
 
Theoremaleph1 8859 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 8860* 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 8861 A nonempty set that is a subset of its union is infinite. This version is proved from ax-ac 8752. See dominf 8738 for a version proved from ax-cc 8728. (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 8862* The countable union of countable sets is countable (indexed union version of unictb 8863). (Contributed by Mario Carneiro, 18-Jan-2014.)
 |-  ( ( A  ~<_  om  /\  A. x  e.  A  B  ~<_  om )  ->  U_ x  e.  A  B  ~<_  om )
 
Theoremunictb 8863* The countable union of countable sets is countable. Theorem 6Q of [Enderton] p. 159. See iunctb 8862 for indexed union version. (Contributed by NM, 26-Mar-2006.)
 |-  ( ( A  ~<_  om  /\  A. x  e.  A  x  ~<_  om )  ->  U. A  ~<_  om )
 
Theoreminfmap 8864* 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 8865 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 8866 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 8867 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 8868* An alternate representation of a successor aleph. Compare alephsuc 8362 and alephsuc2 8374. Equality can be obtained by taking the  card of the right-hand side then using alephcard 8364 and carden 8839. (Contributed by NM, 23-Oct-2004.)
 |-  ( A  e.  On  ->  ( aleph `  suc  A ) 
 ~~  { x  e.  On  |  x  ~~  ( aleph `  A ) } )
 
Theoremalephexp2 8869* An expression equinumerous to 2 to an aleph power. The proof equates the two laws for cardinal exponentiation alephexp1 8867 (which works if the base is less than or equal to the exponent) and infmap 8864 (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 8870 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 8871* A corollary of Konig's Theorem konigth 8857. 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 8872 A corollary of Konig's Theorem konigth 8857. 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 8873 From canth2 7589, 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 8857 (in the form of cfpwsdom 8872), 
( 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 8874 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 8875 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 1-Jan-2002.)
 |-  ( A. x  x  =  y  ->  -.  A. x  y  e.  z
 )
 
Theoremnd2 8876 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 1-Jan-2002.)
 |-  ( A. x  x  =  y  ->  -.  A. x  z  e.  y
 )
 
Theoremnd3 8877 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.)
 |-  ( A. x  x  =  y  ->  -.  A. z  x  e.  y
 )
 
Theoremnd4 8878 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.)
 |-  ( A. x  x  =  y  ->  -.  A. z  y  e.  x )
 
Theoremaxextnd 8879 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 8880* 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 8881 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 8882 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 8883* 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 8884 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 8885 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 8886* Lemma for the Axiom of Power Sets with no distinct variable conditions. Revised to remove a redundant antecedent from the consequence. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Mario Carneiro, 6-Dec-2016.) (Revised and shortened by Wolf Lammen, 9-Jun-2019.)
 |-  ( -.  A. x  x  =  y  ->  ( -.  A. x  x  =  z  ->  E. x A. y ( A. x ( E. z  x  e.  y  ->  A. y  x  e.  z )  ->  y  e.  x )
 ) )
 
Theoremaxpowndlem3 8887* 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.) (Proof shortened by Wolf Lammen, 10-Jun-2019.)
 |-  ( -.  x  =  y  ->  E. x A. y ( A. x ( E. z  x  e.  y  ->  A. y  x  e.  z )  ->  y  e.  x )
 )
 
Theoremaxpowndlem4 8888 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 8889 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 8890 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 8891* 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 8892 A version of the Axiom of Regularity with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.) (Proof shortened by Wolf Lammen, 18-Aug-2019.)
 |-  ( x  e.  y  ->  E. x ( x  e.  y  /\  A. z ( z  e.  x  ->  -.  z  e.  y ) ) )
 
TheoremaxregndOLD 8893 Obsolete proof of axregnd 8892 as of 18-Aug-2019. (Contributed by NM, 3-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( x  e.  y  ->  E. x ( x  e.  y  /\  A. z ( z  e.  x  ->  -.  z  e.  y ) ) )
 
Theoremaxinfndlem1 8894* 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 8895 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 8896 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 8897 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 8898 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 8899* 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 8900* 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 ) )
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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