P. 94 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 9301-9400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ac6num 9301*	A version of ac6 9302 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 9302*	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 9306, 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 9303*	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 9304*	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 9305*	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 9306*	Equivalent of Axiom of Choice. Using the Boundedness Axiom bnd2 8756, we derive this strong version of ac6 9302 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 9307*	Equivalent of Axiom of Choice. Contrapositive of ac6s 9306. (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 9308*	Generalization of the Axiom of Choice to classes. Slightly strengthened version of ac6s3 9309. (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 9309*	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 9310*	ac6s 9306 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 9311*	Version of ac6 9302 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 9312*	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 9313*	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 9314*	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 9315*	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 8754). (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 9316*	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 9317*	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 9318*	Lemma for zorn2 9328. (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 9319*	Lemma for zorn2 9328. (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 9320*	Lemma for zorn2 9328. (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 9321*	Lemma for zorn2 9328. (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 9322*	Lemma for zorn2 9328. (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 9323*	Lemma for zorn2 9328. (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 9324*	Lemma for zorn2 9328. (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 9325*	Zorn's Lemma of [Monk1] p. 117. This version of zorn2 9328 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 9326*	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 9329 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 9327*	Variant of Zorn's lemma zorng 9326 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 9328*	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 9318 through zorn2lem7 9324; this final piece mainly changes bound variables to eliminate the hypotheses of zorn2lem7 9324. (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 9329*	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 9328 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 9330*	Variant of Zorn's lemma zorn 9329 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 9331*	Lemma for ttukey 9340. 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 9332*	Lemma for ttukey 9340. 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 9333*	Lemma for ttukey 9340. (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 9334*	Lemma for ttukey 9340. (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 9335*	Lemma for ttukey 9340. 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 9336*	Lemma for ttukey 9340. (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 9337*	Lemma for ttukey 9340. (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 9338*	The Teichmüller-Tukey Lemma ttukey 9340 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 9339*	The Teichmüller-Tukey Lemma ttukey 9340 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 9340*	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 9341*	Lemma for axdc 9343. (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 9342*	Lemma for axdc 9343. 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 9343*	This theorem derives ax-dc 9268 using ax-ac 9281 and ax-inf 8535. 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 9344	An onto function implies dominance of domain over range. Lemma 10.20 of [Kunen] p. 30. This theorem uses the Axiom of Choice ac7g 9296. AC is not needed for finite sets - see fodomfi 8239. See also fodomnum 8880. (Contributed by NM, 23-Jul-2004.)
|- A e. _V   =>    |- ( F : A -onto-> B -> B ~<_ A )
Theorem	fodomg 9345	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	dmct 9346	The domain of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
|- ( A ~<_ om -> dom A ~<_ om )
Theorem	rnct 9347	The range of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
|- ( A ~<_ om -> ran A ~<_ om )
Theorem	fodomb 9348*	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 9349	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 9350*	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 9351*	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 9352*	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 9353*	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 9354*	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 9355	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 9356	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	fimact 9357	The image by a function of a countable set is countable. (Contributed by Thierry Arnoux, 27-Mar-2018.)
|- ( ( A ~<_ om /\ Fun F ) -> ( F " A ) ~<_ om )
Theorem	fnrndomg 9358	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	fnct 9359	If the domain of a function is countable, the function is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
|- ( ( F Fn A /\ A ~<_ om ) -> F ~<_ om )
Theorem	mptct 9360*	A countable mapping set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
|- ( A ~<_ om -> ( x e. A |-> B ) ~<_ om )
Theorem	iunfo 9361*	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 9362*	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 9363*	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 9364*	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 9365*	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 9366*	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 9367*	An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 9366 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 9368*	The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 8817 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 9369	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 9370	Any set is equinumerous to its cardinal number. Closed theorem form of cardid 9369. (Contributed by David Moews, 1-May-2017.)
|- ( A e. B -> ( card ` A ) ~~ A )
Theorem	cardidd 9371	Any set is equinumerous to its cardinal number. Deduction form of cardid 9369. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. B )   =>    |- ( ph -> ( card ` A ) ~~ A )
Theorem	cardf 9372	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 9373	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 13136 and the finite-set-only hashen 13135. 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 3434). 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 8758). (Contributed by NM, 22-Oct-2003.)
|- ( ( A e. C /\ B e. D ) -> ( ( card ` A ) = ( card ` B ) <-> A ~~ B ) )
Theorem	cardeq0 9374	Only the empty set has cardinality zero. (Contributed by NM, 23-Apr-2004.)
|- ( A e. V -> ( ( card ` A ) = (/) <-> A = (/) ) )
Theorem	unsnen 9375	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 9376	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 9377	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 9378	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 9379	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 9380	Trichotomy of dominance and strict dominance. (Contributed by NM, 4-Jan-2004.)
|- ( ( A e. V /\ B e. W ) -> ( A ~<_ B \/ B ~< A ) )
Theorem	entri3 9381	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 9382	A set strictly dominates iff its cardinal strictly dominates. (Contributed by NM, 30-Oct-2003.)
|- ( A ~< B <-> A ~< ( card ` B ) )
Theorem	canth3 9383	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 9384	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 8837. (Contributed by NM, 17-Sep-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)
|- ( om ~<_ A -> ( A X. A ) ~~ A )
Theorem	ondomon 9385*	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 8449. (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 9386*	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 9387	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 9388	Equivalence between two infiniteness criteria for sets. (Contributed by David Moews, 1-May-2017.)
|- ( A e. B -> ( -. A e. Fin <-> om ~<_ A ) )
Theorem	unirnfdomd 9389	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 9390*	Lemma for konigth 9391. (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 9391*	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 9392	The power set of an aleph dominates the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous, see gch3 9498 or gchaleph2 9494.) (Contributed by NM, 27-Aug-2005.)
|- ( aleph ` suc A ) ~<_ ~P ( aleph ` A )
Theorem	aleph1 9393	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 9394*	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 9395	A nonempty set that is a subset of its union is infinite. This version is proved from ax-ac 9281. See dominf 9267 for a version proved from ax-cc 9257. (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 9396*	The countable union of countable sets is countable (indexed union version of unictb 9397). (Contributed by Mario Carneiro, 18-Jan-2014.)
|- ( ( A ~<_ om /\ A. x e. A B ~<_ om ) -> U_ x e. A B ~<_ om )
Theorem	unictb 9397*	The countable union of countable sets is countable. Theorem 6Q of [Enderton] p. 159. See iunctb 9396 for indexed union version. (Contributed by NM, 26-Mar-2006.)
|- ( ( A ~<_ om /\ A. x e. A x ~<_ om ) -> U. A ~<_ om )
Theorem	infmap 9398*	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 9399	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 ) )
Theorem	alephmul 9400	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 ) ) )