P. 90 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 8901-9000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ac7 8901*	An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49. (Contributed by NM, 29-Apr-2004.)
|- E. f ( f C_ x /\ f Fn dom x )
Theorem	ac7g 8902*	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 ) )
Theorem	ac4 8903*	Equivalent of Axiom of Choice. We do not insist that f be a function. However, theorem ac5 8905, 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 8919. (Contributed by NM, 21-Jul-1996.)
|- E. f A. z e. x ( z =/= (/) -> ( f ` z ) e. z )
Theorem	ac4c 8904*	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 )
Theorem	ac5 8905*	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 8903. (Contributed by NM, 29-Aug-1999.)
|- A e. _V   =>    |- E. f ( f Fn A /\ A. x e. A ( x =/= (/) -> ( f ` x ) e. x ) )
Theorem	ac5b 8906*	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 ) )
Theorem	ac6num 8907*	A version of ac6 8908 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 ) )
Theorem	ac6 8908*	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 8912, 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 ) )
Theorem	ac6c4 8909*	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 ) )
Theorem	ac6c5 8910*	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 )
Theorem	ac9 8911*	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 =/= (/) )
Theorem	ac6s 8912*	Equivalent of Axiom of Choice. Using the Boundedness Axiom bnd2 8363, we derive this strong version of ac6 8908 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 ) )
Theorem	ac6n 8913*	Equivalent of Axiom of Choice. Contrapositive of ac6s 8912. (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 )
Theorem	ac6s2 8914*	Generalization of the Axiom of Choice to classes. Slightly strengthened version of ac6s3 8915. (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 ) )
Theorem	ac6s3 8915*	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 )
Theorem	ac6sg 8916*	ac6s 8912 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 ) ) )
Theorem	ac6sf 8917*	Version of ac6 8908 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 ) )
Theorem	ac6s4 8918*	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 ) )
Theorem	ac6s5 8919*	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 )
Theorem	ac8 8920*	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 ) )
Theorem	ac9s 8921*	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 8361). (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
Theorem	numthcor 8922*	Any set is strictly dominated by some ordinal. (Contributed by NM, 22-Oct-2003.)
|- ( A e. V -> E. x e. On A ~< x )
Theorem	weth 8923*	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 )
Theorem	zorn2lem1 8924*	Lemma for zorn2 8934. (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 )
Theorem	zorn2lem2 8925*	Lemma for zorn2 8934. (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 ) ) )
Theorem	zorn2lem3 8926*	Lemma for zorn2 8934. (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 ) ) )
Theorem	zorn2lem4 8927*	Lemma for zorn2 8934. (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 = (/) )
Theorem	zorn2lem5 8928*	Lemma for zorn2 8934. (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 )
Theorem	zorn2lem6 8929*	Lemma for zorn2 8934. (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 ) ) )
Theorem	zorn2lem7 8930*	Lemma for zorn2 8934. (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 )
Theorem	zorn2g 8931*	Zorn's Lemma of [Monk1] p. 117. This version of zorn2 8934 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 )
Theorem	zorng 8932*	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 8935 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 )
Theorem	zornn0g 8933*	Variant of Zorn's lemma zorng 8932 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 )
Theorem	zorn2 8934*	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 8924 through zorn2lem7 8930; this final piece mainly changes bound variables to eliminate the hypotheses of zorn2lem7 8930. (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 )
Theorem	zorn 8935*	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 8934 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 )
Theorem	zornn0 8936*	Variant of Zorn's lemma zorn 8935 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 )
Theorem	ttukeylem1 8937*	Lemma for ttukey 8946. 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 ) )
Theorem	ttukeylem2 8938*	Lemma for ttukey 8946. 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 )
Theorem	ttukeylem3 8939*	Lemma for ttukey 8946. (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 ) } , (/) ) ) ) )
Theorem	ttukeylem4 8940*	Lemma for ttukey 8946. (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 )
Theorem	ttukeylem5 8941*	Lemma for ttukey 8946. 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 ) )
Theorem	ttukeylem6 8942*	Lemma for ttukey 8946. (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 )
Theorem	ttukeylem7 8943*	Lemma for ttukey 8946. (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 ) )
Theorem	ttukey2g 8944*	The Teichmüller-Tukey Lemma ttukey 8946 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 ) )
Theorem	ttukeyg 8945*	The Teichmüller-Tukey Lemma ttukey 8946 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 )
Theorem	ttukey 8946*	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 )
Theorem	axdclem 8947*	Lemma for axdc 8949. (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 ) ) )
Theorem	axdclem2 8948*	Lemma for axdc 8949. 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 ) ) )
Theorem	axdc 8949*	This theorem derives ax-dc 8874 using ax-ac 8887 and ax-inf 8143. 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 ) )
Theorem	fodom 8950	An onto function implies dominance of domain over range. Lemma 10.20 of [Kunen] p. 30. This theorem uses the Axiom of Choice ac7g 8902. AC is not needed for finite sets - see fodomfi 7856. See also fodomnum 8486. (Contributed by NM, 23-Jul-2004.)
|- A e. _V   =>    |- ( F : A -onto-> B -> B ~<_ A )
Theorem	fodomg 8951	An onto function implies dominance of domain over range. (Contributed by NM, 23-Jul-2004.)
|- ( A e. C -> ( F : A -onto-> B -> B ~<_ A ) )
Theorem	fodomb 8952*	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 ) )
Theorem	wdomac 8953	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 )
Theorem	brdom3 8954*	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 ) )
Theorem	brdom5 8955*	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 ) )
Theorem	brdom4 8956*	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 ) )
Theorem	brdom7disj 8957*	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 ) )
Theorem	brdom6disj 8958*	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 ) )
Theorem	fin71ac 8959	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
Theorem	imadomg 8960	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 ) )
Theorem	fnrndomg 8961	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 ) )
Theorem	iunfo 8962*	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
Theorem	iundom2g 8963*	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 ) )
Theorem	iundomg 8964*	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 ) )
Theorem	iundom 8965*	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 ) )
Theorem	unidom 8966*	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 ) )
Theorem	uniimadom 8967*	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 ) )
Theorem	uniimadomf 8968*	An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 8967 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
Theorem	cardval 8969*	The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 8424 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 }
Theorem	cardid 8970	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
Theorem	cardidg 8971	Any set is equinumerous to its cardinal number. Closed theorem form of cardid 8970. (Contributed by David Moews, 1-May-2017.)
|- ( A e. B -> ( card ` A ) ~~ A )
Theorem	cardidd 8972	Any set is equinumerous to its cardinal number. Deduction form of cardid 8970. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. B )   =>    |- ( ph -> ( card ` A ) ~~ A )
Theorem	cardf 8973	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
Theorem	carden 8974	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 12528 and the finite-set-only hashen 12527. 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 3304). 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 8365). (Contributed by NM, 22-Oct-2003.)
|- ( ( A e. C /\ B e. D ) -> ( ( card ` A ) = ( card ` B ) <-> A ~~ B ) )
Theorem	cardeq0 8975	Only the empty set has cardinality zero. (Contributed by NM, 23-Apr-2004.)
|- ( A e. V -> ( ( card ` A ) = (/) <-> A = (/) ) )
Theorem	unsnen 8976	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 ) )
Theorem	carddom 8977	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 ) )
Theorem	cardsdom 8978	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 ) )
Theorem	domtri 8979	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 ) )
Theorem	entric 8980	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 ) )
Theorem	entri2 8981	Trichotomy of dominance and strict dominance. (Contributed by NM, 4-Jan-2004.)
|- ( ( A e. V /\ B e. W ) -> ( A ~<_ B \/ B ~< A ) )
Theorem	entri3 8982	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 ) )
Theorem	sdomsdomcard 8983	A set strictly dominates iff its cardinal strictly dominates. (Contributed by NM, 30-Oct-2003.)
|- ( A ~< B <-> A ~< ( card ` B ) )
Theorem	canth3 8984	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 ) )
Theorem	infxpidm 8985	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 8444. (Contributed by NM, 17-Sep-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)
|- ( om ~<_ A -> ( A X. A ) ~~ A )
Theorem	ondomon 8986*	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 8059. (Contributed by NM, 7-Nov-2003.) (Proof modification is discouraged.)
|- ( A e. V -> { x e. On | x ~<_ A } e. On )
Theorem	cardmin 8987*	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 } )
Theorem	ficard 8988	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 ) )
Theorem	infinf 8989	Equivalence between two infiniteness criteria for sets. (Contributed by David Moews, 1-May-2017.)
|- ( A e. B -> ( -. A e. Fin <-> om ~<_ A ) )
Theorem	unirnfdomd 8990	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 )
Theorem	konigthlem 8991*	Lemma for konigth 8992. (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 )
Theorem	konigth 8992*	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 )
Theorem	alephsucpw 8993	The power set of an aleph dominates the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous, see gch3 9100 or gchaleph2 9096.) (Contributed by NM, 27-Aug-2005.)
|- ( aleph ` suc A ) ~<_ ~P ( aleph ` A )
Theorem	aleph1 8994	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 ` (/) ) )
Theorem	alephval2 8995*	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 } )
Theorem	dominfac 8996	A nonempty set that is a subset of its union is infinite. This version is proved from ax-ac 8887. See dominf 8873 for a version proved from ax-cc 8863. (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
Theorem	iunctb 8997*	The countable union of countable sets is countable (indexed union version of unictb 8998). (Contributed by Mario Carneiro, 18-Jan-2014.)
|- ( ( A ~<_ om /\ A. x e. A B ~<_ om ) -> U_ x e. A B ~<_ om )
Theorem	unictb 8998*	The countable union of countable sets is countable. Theorem 6Q of [Enderton] p. 159. See iunctb 8997 for indexed union version. (Contributed by NM, 26-Mar-2006.)
|- ( ( A ~<_ om /\ A. x e. A x ~<_ om ) -> U. A ~<_ om )
Theorem	infmap 8999*	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 ) } )
Theorem	alephadd 9000	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 ) )