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Theorem List for Metamath Proof Explorer - 8101-8200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremac7g 8101* An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49. (Contributed by NM, 23-Jul-2004.)
 |-  ( R  e.  A  ->  E. f ( f 
 C_  R  /\  f  Fn  dom  R ) )
 
Theoremac4 8102* Equivalent of Axiom of Choice. We do not insist that  f be a function. However, theorem ac5 8104, derived from this one, shows that this form of the axiom does imply that at least one such set  f whose existence we assert is in fact a function. Axiom of Choice of [TakeutiZaring] p. 83.

Takeuti and Zaring call this "weak choice" in contrast to "strong choice"  E. F A. z
( z  =/=  (/)  ->  ( F `  z )  e.  z ), which asserts the existence of a universal choice function but requires second-order quantification on (proper) class variable  F and thus cannot be expressed in our first-order formalization. However, it has been shown that ZF plus strong choice is a conservative extension of ZF plus weak choice. See Ulrich Felgner, "Comparison of the axioms of local and universal choice," Fundamenta Mathematica, 71, 43-62 (1971).

Weak choice can be strengthened in a different direction to choose from a collection of proper classes; see ac6s5 8118. (Contributed by NM, 21-Jul-1996.)

 |- 
 E. f A. z  e.  x  ( z  =/= 
 (/)  ->  ( f `  z )  e.  z
 )
 
Theoremac4c 8103* Equivalent of Axiom of Choice (class version) (Contributed by NM, 10-Feb-1997.)
 |-  A  e.  _V   =>    |-  E. f A. x  e.  A  ( x  =/=  (/)  ->  ( f `  x )  e.  x )
 
Theoremac5 8104* An Axiom of Choice equivalent: there exists a function  f (called a choice function) with domain 
A that maps each nonempty member of the domain to an element of that member. Axiom AC of [BellMachover] p. 488. Note that the assertion that  f be a function is not necessary; see ac4 8102. (Contributed by NM, 29-Aug-1999.)
 |-  A  e.  _V   =>    |-  E. f ( f  Fn  A  /\  A. x  e.  A  ( x  =/=  (/)  ->  (
 f `  x )  e.  x ) )
 
Theoremac5b 8105* Equivalent of Axiom of Choice. (Contributed by NM, 31-Aug-1999.)
 |-  A  e.  _V   =>    |-  ( A. x  e.  A  x  =/=  (/)  ->  E. f
 ( f : A --> U. A  /\  A. x  e.  A  ( f `  x )  e.  x ) )
 
Theoremac6num 8106* A version of ac6 8107 which takes the choice as a hypothesis. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  ( y  =  ( f `  x ) 
 ->  ( ph  <->  ps ) )   =>    |-  ( ( A  e.  V  /\  U_ x  e.  A  { y  e.  B  |  ph }  e.  dom  card  /\  A. x  e.  A  E. y  e.  B  ph )  ->  E. f ( f : A --> B  /\  A. x  e.  A  ps ) )
 
Theoremac6 8107* Equivalent of Axiom of Choice. This is useful for proving that there exists, for example, a sequence mapping natural numbers to members of a larger set  B, where  ph depends on  x (the natural number) and  y (to specify a member of  B). A stronger version of this theorem, ac6s 8111, allows  B to be a proper class. (Contributed by NM, 18-Oct-1999.) (Revised by Mario Carneiro, 27-Aug-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( y  =  ( f `  x )  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  e.  A  E. y  e.  B  ph  ->  E. f
 ( f : A --> B  /\  A. x  e.  A  ps ) )
 
Theoremac6c4 8108* Equivalent of Axiom of Choice.  B is a collection  B ( x ) of nonempty sets. (Contributed by Mario Carneiro, 22-Mar-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A. x  e.  A  B  =/=  (/)  ->  E. f
 ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B )
 )
 
Theoremac6c5 8109* Equivalent of Axiom of Choice.  B is a collection  B ( x ) of nonempty sets. Remark after Theorem 10.46 of [TakeutiZaring] p. 98. (Contributed by Mario Carneiro, 22-Mar-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A. x  e.  A  B  =/=  (/)  ->  E. f A. x  e.  A  ( f `  x )  e.  B )
 
Theoremac9 8110* An Axiom of Choice equivalent: the infinite Cartesian product of nonempty classes is nonempty. Axiom of Choice (second form) of [Enderton] p. 55 and its converse. (Contributed by Mario Carneiro, 22-Mar-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A. x  e.  A  B  =/=  (/)  <->  X_ x  e.  A  B  =/=  (/) )
 
Theoremac6s 8111* Equivalent of Axiom of Choice. Using the Boundedness Axiom bnd2 7563, we derive this strong version of ac6 8107 that doesn't require  B to be a set. (Contributed by NM, 4-Feb-2004.)
 |-  A  e.  _V   &    |-  (
 y  =  ( f `
  x )  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  e.  A  E. y  e.  B  ph  ->  E. f
 ( f : A --> B  /\  A. x  e.  A  ps ) )
 
Theoremac6n 8112* Equivalent of Axiom of Choice. Contrapositive of ac6s 8111. (Contributed by NM, 10-Jun-2007.)
 |-  A  e.  _V   &    |-  (
 y  =  ( f `
  x )  ->  ( ph  <->  ps ) )   =>    |-  ( A. f
 ( f : A --> B  ->  E. x  e.  A  ps )  ->  E. x  e.  A  A. y  e.  B  ph )
 
Theoremac6s2 8113* Generalization of the Axiom of Choice to classes. Slightly strengthened version of ac6s3 8114. (Contributed by NM, 29-Sep-2006.)
 |-  A  e.  _V   &    |-  (
 y  =  ( f `
  x )  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  e.  A  E. y ph  ->  E. f ( f  Fn  A  /\  A. x  e.  A  ps ) )
 
Theoremac6s3 8114* Generalization of the Axiom of Choice to classes. Theorem 10.46 of [TakeutiZaring] p. 97. (Contributed by NM, 3-Nov-2004.)
 |-  A  e.  _V   &    |-  (
 y  =  ( f `
  x )  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  e.  A  E. y ph  ->  E. f A. x  e.  A  ps )
 
Theoremac6sg 8115* ac6s 8111 with sethood as antecedent. (Contributed by FL, 3-Aug-2009.)
 |-  ( y  =  ( f `  x ) 
 ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  V  ->  ( A. x  e.  A  E. y  e.  B  ph  ->  E. f
 ( f : A --> B  /\  A. x  e.  A  ps ) ) )
 
Theoremac6sf 8116* Version of ac6 8107 with bound-variable hypothesis. (Contributed by NM, 2-Mar-2008.)
 |- 
 F/ y ps   &    |-  A  e.  _V   &    |-  ( y  =  ( f `  x )  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  e.  A  E. y  e.  B  ph  ->  E. f
 ( f : A --> B  /\  A. x  e.  A  ps ) )
 
Theoremac6s4 8117* Generalization of the Axiom of Choice to proper classes.  B is a collection  B ( x ) of nonempty, possible proper classes. (Contributed by NM, 29-Sep-2006.)
 |-  A  e.  _V   =>    |-  ( A. x  e.  A  B  =/=  (/)  ->  E. f
 ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B )
 )
 
Theoremac6s5 8118* Generalization of the Axiom of Choice to proper classes.  B is a collection  B ( x ) of nonempty, possible proper classes. Remark after Theorem 10.46 of [TakeutiZaring] p. 98. (Contributed by NM, 27-Mar-2006.)
 |-  A  e.  _V   =>    |-  ( A. x  e.  A  B  =/=  (/)  ->  E. f A. x  e.  A  ( f `  x )  e.  B )
 
Theoremac8 8119* An Axiom of Choice equivalent. Given a family  x of mutually disjoint nonempty sets, there exists a set  y containing exactly one member from each set in the family. Theorem 6M(4) of [Enderton] p. 151. (Contributed by NM, 14-May-2004.)
 |-  ( ( A. z  e.  x  z  =/=  (/)  /\  A. z  e.  x  A. w  e.  x  ( z  =/=  w  ->  ( z  i^i  w )  =  (/) ) )  ->  E. y A. z  e.  x  E! v  v  e.  ( z  i^i  y ) )
 
Theoremac9s 8120* An Axiom of Choice equivalent: the infinite Cartesian product of nonempty classes is nonempty. Axiom of Choice (second form) of [Enderton] p. 55 and its converse. This is a stronger version of the axiom in Enderton, with no existence requirement for the family of classes  B ( x ) (achieved via the Collection Principle cp 7561). (Contributed by NM, 29-Sep-2006.)
 |-  A  e.  _V   =>    |-  ( A. x  e.  A  B  =/=  (/)  <->  X_ x  e.  A  B  =/=  (/) )
 
3.2.2  AC equivalents: well ordering, Zorn's lemma
 
Theoremnumthcor 8121* Any set is strictly dominated by some ordinal. (Contributed by NM, 22-Oct-2003.)
 |-  ( A  e.  V  ->  E. x  e.  On  A  ~<  x )
 
Theoremweth 8122* Well-ordering theorem: any set  A can be well-ordered. This is an equivalent of the Axiom of Choice. Theorem 6 of [Suppes] p. 242. First proved by Ernst Zermelo (the "Z" in ZFC) in 1904. (Contributed by Mario Carneiro, 5-Jan-2013.)
 |-  ( A  e.  V  ->  E. x  x  We  A )
 
Theoremzorn2lem1 8123* Lemma for zorn2 8133. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  F  = recs ( ( f  e.  _V  |->  (
 iota_ v  e.  C A. u  e.  C  -.  u w v ) ) )   &    |-  C  =  {
 z  e.  A  |  A. g  e.  ran  f  g R z }   &    |-  D  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z }   =>    |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/= 
 (/) ) )  ->  ( F `  x )  e.  D )
 
Theoremzorn2lem2 8124* Lemma for zorn2 8133. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  F  = recs ( ( f  e.  _V  |->  (
 iota_ v  e.  C A. u  e.  C  -.  u w v ) ) )   &    |-  C  =  {
 z  e.  A  |  A. g  e.  ran  f  g R z }   &    |-  D  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z }   =>    |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/= 
 (/) ) )  ->  ( y  e.  x  ->  ( F `  y
 ) R ( F `
  x ) ) )
 
Theoremzorn2lem3 8125* Lemma for zorn2 8133. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  F  = recs ( ( f  e.  _V  |->  (
 iota_ v  e.  C A. u  e.  C  -.  u w v ) ) )   &    |-  C  =  {
 z  e.  A  |  A. g  e.  ran  f  g R z }   &    |-  D  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z }   =>    |-  ( ( R  Po  A  /\  ( x  e.  On  /\  ( w  We  A  /\  D  =/= 
 (/) ) ) ) 
 ->  ( y  e.  x  ->  -.  ( F `  x )  =  ( F `  y ) ) )
 
Theoremzorn2lem4 8126* Lemma for zorn2 8133. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  F  = recs ( ( f  e.  _V  |->  (
 iota_ v  e.  C A. u  e.  C  -.  u w v ) ) )   &    |-  C  =  {
 z  e.  A  |  A. g  e.  ran  f  g R z }   &    |-  D  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z }   =>    |-  ( ( R  Po  A  /\  w  We  A )  ->  E. x  e.  On  D  =  (/) )
 
Theoremzorn2lem5 8127* Lemma for zorn2 8133. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  F  = recs ( ( f  e.  _V  |->  (
 iota_ v  e.  C A. u  e.  C  -.  u w v ) ) )   &    |-  C  =  {
 z  e.  A  |  A. g  e.  ran  f  g R z }   &    |-  D  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z }   &    |-  H  =  { z  e.  A  |  A. g  e.  ( F " y ) g R z }   =>    |-  ( ( ( w  We  A  /\  x  e.  On )  /\  A. y  e.  x  H  =/=  (/) )  ->  ( F " x )  C_  A )
 
Theoremzorn2lem6 8128* Lemma for zorn2 8133. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  F  = recs ( ( f  e.  _V  |->  (
 iota_ v  e.  C A. u  e.  C  -.  u w v ) ) )   &    |-  C  =  {
 z  e.  A  |  A. g  e.  ran  f  g R z }   &    |-  D  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z }   &    |-  H  =  { z  e.  A  |  A. g  e.  ( F " y ) g R z }   =>    |-  ( R  Po  A  ->  ( ( ( w  We  A  /\  x  e.  On )  /\  A. y  e.  x  H  =/=  (/) )  ->  R  Or  ( F " x ) ) )
 
Theoremzorn2lem7 8129* Lemma for zorn2 8133. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  F  = recs ( ( f  e.  _V  |->  (
 iota_ v  e.  C A. u  e.  C  -.  u w v ) ) )   &    |-  C  =  {
 z  e.  A  |  A. g  e.  ran  f  g R z }   &    |-  D  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z }   &    |-  H  =  { z  e.  A  |  A. g  e.  ( F " y ) g R z }   =>    |-  ( ( A  e.  dom  card  /\  R  Po  A  /\  A. s
 ( ( s  C_  A  /\  R  Or  s
 )  ->  E. a  e.  A  A. r  e.  s  ( r R a  \/  r  =  a ) ) ) 
 ->  E. a  e.  A  A. b  e.  A  -.  a R b )
 
Theoremzorn2g 8130* Zorn's Lemma of [Monk1] p. 117. This version of zorn2 8133 avoids the Axiom of Choice by assuming that  A is well-orderable. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  ( ( A  e.  dom  card  /\  R  Po  A  /\  A. w ( ( w  C_  A  /\  R  Or  w )  ->  E. x  e.  A  A. z  e.  w  ( z R x  \/  z  =  x )
 ) )  ->  E. x  e.  A  A. y  e.  A  -.  x R y )
 
Theoremzorng 8131* Zorn's Lemma. If the union of every chain (with respect to inclusion) in a set belongs to the set, then the set contains a maximal element. Theorem 6M of [Enderton] p. 151. This version of zorn 8134 avoids the Axiom of Choice by assuming that  A is well-orderable. (Contributed by NM, 12-Aug-2004.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  ( ( A  e.  dom  card  /\  A. z ( ( z  C_  A  /\ [ C.]  Or  z ) 
 ->  U. z  e.  A ) )  ->  E. x  e.  A  A. y  e.  A  -.  x  C.  y )
 
Theoremzornn0g 8132* Variant of Zorn's lemma zorng 8131 in which  (/), the union of the empty chain, is not required to be an element of  A. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  ( ( A  e.  dom  card  /\  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 )
 
Theoremzorn2 8133* Zorn's Lemma of [Monk1] p. 117. This theorem is equivalent to the Axiom of Choice and states that every partially ordered set  A (with an ordering relation  R) in which every totally ordered subset has an upper bound, contains at least one maximal element. The main proof consists of lemmas zorn2lem1 8123 through zorn2lem7 8129; this final piece mainly changes bound variables to eliminate the hypotheses of zorn2lem7 8129. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  A  e.  _V   =>    |-  ( ( R  Po  A  /\  A. w ( ( w 
 C_  A  /\  R  Or  w )  ->  E. x  e.  A  A. z  e.  w  ( z R x  \/  z  =  x ) ) ) 
 ->  E. x  e.  A  A. y  e.  A  -.  x R y )
 
Theoremzorn 8134* Zorn's Lemma. If the union of every chain (with respect to inclusion) in a set belongs to the set, then the set contains a maximal element. This theorem is equivalent to the Axiom of Choice. Theorem 6M of [Enderton] p. 151. See zorn2 8133 for a version with general partial orderings. (Contributed by NM, 12-Aug-2004.)
 |-  A  e.  _V   =>    |-  ( A. z
 ( ( z  C_  A  /\ [ C.]  Or  z
 )  ->  U. z  e.  A )  ->  E. x  e.  A  A. y  e.  A  -.  x  C.  y )
 
Theoremzornn0 8135* Variant of Zorn's lemma zorn 8134 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 8136* Lemma for ttukey 8145. 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 8137* Lemma for ttukey 8145. 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 8138* Lemma for ttukey 8145. (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 8139* Lemma for ttukey 8145. (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 8140* Lemma for ttukey 8145. 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 8141* Lemma for ttukey 8145. (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 8142* Lemma for ttukey 8145. (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 8143* The Teichmüller-Tukey Lemma ttukey 8145 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 8144* The Teichmüller-Tukey Lemma ttukey 8145 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 8145* 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 8146* Lemma for axdc 8148. (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 8147* Lemma for axdc 8148. 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 8148* This theorem derives ax-dc 8072 using ax-ac 8085 and ax-inf 7339. 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 8149 An onto function implies dominance of domain over range. Lemma 10.20 of [Kunen] p. 30. This theorem uses the Axiom of Choice ac7g 8101. AC is not needed for finite sets - see fodomfi 7135. See also fodomnum 7684. (Contributed by NM, 23-Jul-2004.)
 |-  A  e.  _V   =>    |-  ( F : A -onto-> B  ->  B  ~<_  A )
 
Theoremfodomg 8150 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 8151* 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 8152 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 8153* 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 8154* 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 8155* 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 8156* 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 8157* 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 8158 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 8159 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 8160 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 8161* 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 8162* 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 8163* 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 8164* 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 8165* 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 8166* 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 8167* An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 8166 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 8168* The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 7624 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 8169 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 8170 Any set is equinumerous to its cardinal number. Closed theorem form of cardid 8169. (Contributed by David Moews, 1-May-2017.)
 |-  ( A  e.  B  ->  ( card `  A )  ~~  A )
 
Theoremcardidd 8171 Any set is equinumerous to its cardinal number. Deduction form of cardid 8169. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  B )   =>    |-  ( ph  ->  ( card `  A )  ~~  A )
 
Theoremcardf 8172 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 8173 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 7565). (Contributed by NM, 22-Oct-2003.)

 |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( card `  A )  =  (
 card `  B )  <->  A  ~~  B ) )
 
Theoremcardeq0 8174 Only the empty set has cardinality zero. (Contributed by NM, 23-Apr-2004.)
 |-  ( A  e.  V  ->  ( ( card `  A )  =  (/)  <->  A  =  (/) ) )
 
Theoremunsnen 8175 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 8176 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 8177 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 8178 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 8179 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 8180 Trichotomy of dominance and strict dominance. (Contributed by NM, 4-Jan-2004.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  ~<_  B  \/  B  ~<  A ) )
 
Theorementri3 8181 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 8182 A set strictly dominates iff its cardinal strictly dominates. (Contributed by NM, 30-Oct-2003.)
 |-  ( A  ~<  B  <->  A  ~<  ( card `  B ) )
 
Theoremcanth3 8183 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 8184 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 7642. (Contributed by NM, 17-Sep-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)
 |-  ( om  ~<_  A  ->  ( A  X.  A ) 
 ~~  A )
 
Theoremondomon 8185* 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 7259. (Contributed by NM, 7-Nov-2003.) (Proof modification is discouraged.)
 |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<_  A }  e.  On )
 
Theoremcardmin 8186* 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 8187 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 8188 Equivalence between two infiniteness criteria for sets. (Contributed by David Moews, 1-May-2017.)
 |-  ( A  e.  B  ->  ( -.  A  e.  Fin  <->  om  ~<_  A ) )
 
Theoremunirnfdomd 8189 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 8190* Lemma for konigth 8191. (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 8191* 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 8192 The power set of an aleph dominates the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous, see gch3 8302 or gchaleph2 8298.) (Contributed by NM, 27-Aug-2005.)
 |-  ( aleph `  suc  A )  ~<_  ~P ( aleph `  A )
 
Theoremaleph1 8193 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 8194* 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 8195 A nonempty set that is a subset of its union is infinite. This version is proved from ax-ac 8085. See dominf 8071 for a version proved from ax-cc 8061. (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 8196* The countable union of countable sets is countable (indexed union version of unictb 8197). (Contributed by Mario Carneiro, 18-Jan-2014.)
 |-  ( ( A  ~<_  om  /\  A. x  e.  A  B  ~<_  om )  ->  U_ x  e.  A  B  ~<_  om )
 
Theoremunictb 8197* The countable union of countable sets is countable. Theorem 6Q of [Enderton] p. 159. See iunctb 8196 for indexed union version. (Contributed by NM, 26-Mar-2006.)
 |-  ( ( A  ~<_  om  /\  A. x  e.  A  x  ~<_  om )  ->  U. A  ~<_  om )
 
Theoreminfmap 8198* 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 8199 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 8200 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 ) ) )
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