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	axac3 8901	This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 8900 as an axiom. (Contributed by Mario Carneiro, 6-May-2015.) (Revised by NM, 20-Dec-2016.) (Proof modification is discouraged.)
|- CHOICE
Theorem	ackm 8902*	A remarkable equivalent to the Axiom of Choice that has only five quantifiers (when expanded to e., = primitives in prenex form), discovered and proved by Kurt Maes. This establishes a new record, reducing from 6 to 5 the largest number of quantified variables needed by any ZFC axiom. The ZF-equivalence to AC is shown by theorem dfackm 8603. Maes found this version of AC in April, 2004 (replacing a longer version, also with five quantifiers, that he found in November, 2003). See Kurt Maes, "A 5-quantifier ( e.,=)-expression ZF-equivalent to the Axiom of Choice" (http://arxiv.org/PS_cache/arxiv/pdf/0705/0705.3162v1.pdf). The original FOM posts are: http://www.cs.nyu.edu/pipermail/fom/2003-November/007631.html http://www.cs.nyu.edu/pipermail/fom/2003-November/007641.html. (Contributed by NM, 29-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.) (Proof modification is discouraged.)
|- A. x E. y A. z E. v A. u ( ( y e. x /\ ( z e. y -> ( ( v e. x /\ -. y = v ) /\ z e. v ) ) ) \/ ( -. y e. x /\ ( z e. x -> ( ( v e. z /\ v e. y ) /\ ( ( u e. z /\ u e. y ) -> u = v ) ) ) ) )
Theorem	axac2 8903*	Derive ax-ac2 8900 from ax-ac 8896. (Contributed by NM, 19-Dec-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
|- E. y A. z E. v A. u ( ( y e. x /\ ( z e. y -> ( ( v e. x /\ -. y = v ) /\ z e. v ) ) ) \/ ( -. y e. x /\ ( z e. x -> ( ( v e. z /\ v e. y ) /\ ( ( u e. z /\ u e. y ) -> u = v ) ) ) ) )
Theorem	axac 8904*	Derive ax-ac 8896 from ax-ac2 8900. Note that ax-reg 8116 is used by the proof. (Contributed by NM, 19-Dec-2016.) (Proof modification is discouraged.)
|- E. y A. z A. w ( ( z e. w /\ w e. x ) -> E. v A. u ( E. t ( ( u e. w /\ w e. t ) /\ ( u e. t /\ t e. y ) ) <-> u = v ) )
Theorem	axaci 8905	Apply a choice equivalent. (Contributed by Mario Carneiro, 17-May-2015.)
|- (CHOICE <-> A. x ph )   =>    |- ph
Theorem	cardeqv 8906	All sets are well-orderable under choice. (Contributed by Mario Carneiro, 28-Apr-2015.)
|- dom card = _V
Theorem	numth3 8907	All sets are well-orderable under choice. (Contributed by Stefan O'Rear, 28-Feb-2015.)
|- ( A e. V -> A e. dom card )
Theorem	numth2 8908*	Numeration theorem: any set is equinumerous to some ordinal (using AC). Theorem 10.3 of [TakeutiZaring] p. 84. (Contributed by NM, 20-Oct-2003.)
|- A e. _V   =>    |- E. x e. On x ~~ A
Theorem	numth 8909*	Numeration theorem: every set can be put into one-to-one correspondence with some ordinal (using AC). Theorem 10.3 of [TakeutiZaring] p. 84. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Mario Carneiro, 8-Jan-2015.)
|- A e. _V   =>    |- E. x e. On E. f f : x -1-1-onto-> A
Theorem	ac7 8910*	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 8911*	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 8912*	Equivalent of Axiom of Choice. We do not insist that f be a function. However, theorem ac5 8914, 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 8928. (Contributed by NM, 21-Jul-1996.)
|- E. f A. z e. x ( z =/= (/) -> ( f ` z ) e. z )
Theorem	ac4c 8913*	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 8914*	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 8912. (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 8915*	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 8916*	A version of ac6 8917 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 8917*	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 8921, 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 8918*	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 8919*	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 8920*	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 8921*	Equivalent of Axiom of Choice. Using the Boundedness Axiom bnd2 8372, we derive this strong version of ac6 8917 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 8922*	Equivalent of Axiom of Choice. Contrapositive of ac6s 8921. (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 8923*	Generalization of the Axiom of Choice to classes. Slightly strengthened version of ac6s3 8924. (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 8924*	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 8925*	ac6s 8921 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 8926*	Version of ac6 8917 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 8927*	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 8928*	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 8929*	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 8930*	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 8370). (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 8931*	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 8932*	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 8933*	Lemma for zorn2 8943. (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 8934*	Lemma for zorn2 8943. (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 8935*	Lemma for zorn2 8943. (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 8936*	Lemma for zorn2 8943. (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 8937*	Lemma for zorn2 8943. (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 8938*	Lemma for zorn2 8943. (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 8939*	Lemma for zorn2 8943. (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 8940*	Zorn's Lemma of [Monk1] p. 117. This version of zorn2 8943 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 8941*	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 8944 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 8942*	Variant of Zorn's lemma zorng 8941 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 8943*	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 8933 through zorn2lem7 8939; this final piece mainly changes bound variables to eliminate the hypotheses of zorn2lem7 8939. (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 8944*	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 8943 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 8945*	Variant of Zorn's lemma zorn 8944 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 8946*	Lemma for ttukey 8955. 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 8947*	Lemma for ttukey 8955. 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 8948*	Lemma for ttukey 8955. (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 8949*	Lemma for ttukey 8955. (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 8950*	Lemma for ttukey 8955. 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 8951*	Lemma for ttukey 8955. (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 8952*	Lemma for ttukey 8955. (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 8953*	The Teichmüller-Tukey Lemma ttukey 8955 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 8954*	The Teichmüller-Tukey Lemma ttukey 8955 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 8955*	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 8956*	Lemma for axdc 8958. (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 8957*	Lemma for axdc 8958. 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 8958*	This theorem derives ax-dc 8883 using ax-ac 8896 and ax-inf 8152. 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 8959	An onto function implies dominance of domain over range. Lemma 10.20 of [Kunen] p. 30. This theorem uses the Axiom of Choice ac7g 8911. AC is not needed for finite sets - see fodomfi 7859. See also fodomnum 8495. (Contributed by NM, 23-Jul-2004.)
|- A e. _V   =>    |- ( F : A -onto-> B -> B ~<_ A )
Theorem	fodomg 8960	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 8961*	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 8962	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 8963*	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 8964*	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 8965*	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 8966*	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 8967*	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 8968	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 8969	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 8970	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 8971*	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 8972*	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 8973*	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 8974*	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 8975*	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 8976*	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 8977*	An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 8976 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 8978*	The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 8433 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 8979	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 8980	Any set is equinumerous to its cardinal number. Closed theorem form of cardid 8979. (Contributed by David Moews, 1-May-2017.)
|- ( A e. B -> ( card ` A ) ~~ A )
Theorem	cardidd 8981	Any set is equinumerous to its cardinal number. Deduction form of cardid 8979. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. B )   =>    |- ( ph -> ( card ` A ) ~~ A )
Theorem	cardf 8982	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 8983	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 12537 and the finite-set-only hashen 12536. 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 3298). 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 8374). (Contributed by NM, 22-Oct-2003.)
|- ( ( A e. C /\ B e. D ) -> ( ( card ` A ) = ( card ` B ) <-> A ~~ B ) )
Theorem	cardeq0 8984	Only the empty set has cardinality zero. (Contributed by NM, 23-Apr-2004.)
|- ( A e. V -> ( ( card ` A ) = (/) <-> A = (/) ) )
Theorem	unsnen 8985	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 8986	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 8987	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 8988	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 8989	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 8990	Trichotomy of dominance and strict dominance. (Contributed by NM, 4-Jan-2004.)
|- ( ( A e. V /\ B e. W ) -> ( A ~<_ B \/ B ~< A ) )
Theorem	entri3 8991	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 8992	A set strictly dominates iff its cardinal strictly dominates. (Contributed by NM, 30-Oct-2003.)
|- ( A ~< B <-> A ~< ( card ` B ) )
Theorem	canth3 8993	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 8994	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 8453. (Contributed by NM, 17-Sep-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)
|- ( om ~<_ A -> ( A X. A ) ~~ A )
Theorem	ondomon 8995*	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 8068. (Contributed by NM, 7-Nov-2003.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( A e. V -> { x e. On | x ~<_ A } e. On )
Theorem	cardmin 8996*	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 8997	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 8998	Equivalence between two infiniteness criteria for sets. (Contributed by David Moews, 1-May-2017.)
|- ( A e. B -> ( -. A e. Fin <-> om ~<_ A ) )
Theorem	unirnfdomd 8999	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 9000*	Lemma for konigth 9001. (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 )