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	ttukeylem1 8901*	Lemma for ttukey 8910. 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 8902*	Lemma for ttukey 8910. 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 8903*	Lemma for ttukey 8910. (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 8904*	Lemma for ttukey 8910. (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 8905*	Lemma for ttukey 8910. 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 8906*	Lemma for ttukey 8910. (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 8907*	Lemma for ttukey 8910. (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 8908*	The Teichmüller-Tukey Lemma ttukey 8910 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 8909*	The Teichmüller-Tukey Lemma ttukey 8910 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 8910*	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 8911*	Lemma for axdc 8913. (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 8912*	Lemma for axdc 8913. 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 8913*	This theorem derives ax-dc 8838 using ax-ac 8851 and ax-inf 8067. 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 8914	An onto function implies dominance of domain over range. Lemma 10.20 of [Kunen] p. 30. This theorem uses the Axiom of Choice ac7g 8866. AC is not needed for finite sets - see fodomfi 7811. See also fodomnum 8450. (Contributed by NM, 23-Jul-2004.)
|- A e. _V   =>    |- ( F : A -onto-> B -> B ~<_ A )
Theorem	fodomg 8915	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 8916*	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 8917	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 8918*	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 8919*	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 8920*	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 8921*	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 8922*	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 8923	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 8924	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 8925	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 8926*	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 8927*	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 8928*	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 8929*	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 8930*	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 8931*	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 8932*	An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 8931 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 8933*	The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 8384 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 8934	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 8935	Any set is equinumerous to its cardinal number. Closed theorem form of cardid 8934. (Contributed by David Moews, 1-May-2017.)
|- ( A e. B -> ( card ` A ) ~~ A )
Theorem	cardidd 8936	Any set is equinumerous to its cardinal number. Deduction form of cardid 8934. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. B )   =>    |- ( ph -> ( card ` A ) ~~ A )
Theorem	cardf 8937	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 8938	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 12401 and the finite-set-only hashen 12400. 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 3335). 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 8325). (Contributed by NM, 22-Oct-2003.)
|- ( ( A e. C /\ B e. D ) -> ( ( card ` A ) = ( card ` B ) <-> A ~~ B ) )
Theorem	cardeq0 8939	Only the empty set has cardinality zero. (Contributed by NM, 23-Apr-2004.)
|- ( A e. V -> ( ( card ` A ) = (/) <-> A = (/) ) )
Theorem	unsnen 8940	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 8941	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 8942	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 8943	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 8944	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 8945	Trichotomy of dominance and strict dominance. (Contributed by NM, 4-Jan-2004.)
|- ( ( A e. V /\ B e. W ) -> ( A ~<_ B \/ B ~< A ) )
Theorem	entri3 8946	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 8947	A set strictly dominates iff its cardinal strictly dominates. (Contributed by NM, 30-Oct-2003.)
|- ( A ~< B <-> A ~< ( card ` B ) )
Theorem	canth3 8948	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 8949	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 8404. (Contributed by NM, 17-Sep-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)
|- ( om ~<_ A -> ( A X. A ) ~~ A )
Theorem	ondomon 8950*	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 7981. (Contributed by NM, 7-Nov-2003.) (Proof modification is discouraged.)
|- ( A e. V -> { x e. On | x ~<_ A } e. On )
Theorem	cardmin 8951*	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 8952	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 8953	Equivalence between two infiniteness criteria for sets. (Contributed by David Moews, 1-May-2017.)
|- ( A e. B -> ( -. A e. Fin <-> om ~<_ A ) )
Theorem	unirnfdomd 8954	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 8955*	Lemma for konigth 8956. (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 8956*	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 8957	The power set of an aleph dominates the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous, see gch3 9066 or gchaleph2 9062.) (Contributed by NM, 27-Aug-2005.)
|- ( aleph ` suc A ) ~<_ ~P ( aleph ` A )
Theorem	aleph1 8958	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 8959*	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 8960	A nonempty set that is a subset of its union is infinite. This version is proved from ax-ac 8851. See dominf 8837 for a version proved from ax-cc 8827. (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 8961*	The countable union of countable sets is countable (indexed union version of unictb 8962). (Contributed by Mario Carneiro, 18-Jan-2014.)
|- ( ( A ~<_ om /\ A. x e. A B ~<_ om ) -> U_ x e. A B ~<_ om )
Theorem	unictb 8962*	The countable union of countable sets is countable. Theorem 6Q of [Enderton] p. 159. See iunctb 8961 for indexed union version. (Contributed by NM, 26-Mar-2006.)
|- ( ( A ~<_ om /\ A. x e. A x ~<_ om ) -> U. A ~<_ om )
Theorem	infmap 8963*	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 8964	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 8965	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 ) ) )
Theorem	alephexp1 8966	An exponentiation law for alephs. Lemma 6.1 of [Jech] p. 42. (Contributed by NM, 29-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
|- ( ( ( A e. On /\ B e. On ) /\ A C_ B ) -> ( ( aleph ` A ) ^m ( aleph ` B ) ) ~~ ( 2o ^m ( aleph ` B ) ) )
Theorem	alephsuc3 8967*	An alternate representation of a successor aleph. Compare alephsuc 8461 and alephsuc2 8473. Equality can be obtained by taking the card of the right-hand side then using alephcard 8463 and carden 8938. (Contributed by NM, 23-Oct-2004.)
|- ( A e. On -> ( aleph ` suc A ) ~~ { x e. On | x ~~ ( aleph ` A ) } )
Theorem	alephexp2 8968*	An expression equinumerous to 2 to an aleph power. The proof equates the two laws for cardinal exponentiation alephexp1 8966 (which works if the base is less than or equal to the exponent) and infmap 8963 (which works if the exponent is less than or equal to the base). They can be equated only when the base is equal to the exponent, and this is the result. (Contributed by NM, 23-Oct-2004.)
|- ( A e. On -> ( 2o ^m ( aleph ` A ) ) ~~ { x | ( x C_ ( aleph ` A ) /\ x ~~ ( aleph ` A ) ) } )
3.2.5  Cofinality using Axiom of Choice
Theorem	alephreg 8969	A successor aleph is regular. Theorem 11.15 of [TakeutiZaring] p. 103. (Contributed by Mario Carneiro, 9-Mar-2013.)
|- ( cf ` ( aleph ` suc A ) ) = ( aleph ` suc A )
Theorem	pwcfsdom 8970*	A corollary of Konig's Theorem konigth 8956. Theorem 11.28 of [TakeutiZaring] p. 108. (Contributed by Mario Carneiro, 20-Mar-2013.)
|- H = ( y e. ( cf ` ( aleph ` A ) ) |-> (har ` ( f ` y ) ) )   =>    |- ( aleph ` A ) ~< ( ( aleph ` A ) ^m ( cf ` ( aleph ` A ) ) )
Theorem	cfpwsdom 8971	A corollary of Konig's Theorem konigth 8956. Theorem 11.29 of [TakeutiZaring] p. 108. (Contributed by Mario Carneiro, 20-Mar-2013.)
|- B e. _V   =>    |- ( 2o ~<_ B -> ( aleph ` A ) ~< ( cf ` ( card ` ( B ^m ( aleph ` A ) ) ) ) )
Theorem	alephom 8972	From canth2 7682, we know that ( aleph ` 0 ) < ( 2 ^ om ), but we cannot prove that ( 2 ^ om ) = ( aleph ` 1 ) (this is the Continuum Hypothesis), nor can we prove that it is less than any bound whatsoever (i.e. the statement ( aleph ` A ) < ( 2 ^ om ) is consistent for any ordinal A). However, we can prove that ( 2 ^ om ) is not equal to ( aleph ` om ), nor ( aleph ` ( aleph ` om ) ), on cofinality grounds, because by Konig's Theorem konigth 8956 (in the form of cfpwsdom 8971), ( 2 ^ om ) has uncountable cofinality, which eliminates limit alephs like ( aleph ` om ). (The first limit aleph that is not eliminated is ( aleph ` ( aleph ` 1 ) ), which has cofinality ( aleph ` 1 ).) (Contributed by Mario Carneiro, 21-Mar-2013.)
|- ( card ` ( 2o ^m om ) ) =/= ( aleph ` om )
Theorem	smobeth 8973	The beth function is strictly monotone. This function is not strictly the beth function, but rather bethA is the same as ( card ` ( R1 ` ( om +o A ) ) ), since conventionally we start counting at the first infinite level, and ignore the finite levels. (Contributed by Mario Carneiro, 6-Jun-2013.) (Revised by Mario Carneiro, 2-Jun-2015.)
|- Smo ( card o. R1 )
3.3  ZFC Axioms with no distinct variable requirements
Theorem	nd1 8974	A lemma for proving conditionless ZFC axioms. (Contributed by NM, 1-Jan-2002.)
|- ( A. x x = y -> -. A. x y e. z )
Theorem	nd2 8975	A lemma for proving conditionless ZFC axioms. (Contributed by NM, 1-Jan-2002.)
|- ( A. x x = y -> -. A. x z e. y )
Theorem	nd3 8976	A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.)
|- ( A. x x = y -> -. A. z x e. y )
Theorem	nd4 8977	A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.)
|- ( A. x x = y -> -. A. z y e. x )
Theorem	axextnd 8978	A version of the Axiom of Extensionality with no distinct variable conditions. (Contributed by NM, 14-Aug-2003.)
|- E. x ( ( x e. y <-> x e. z ) -> y = z )
Theorem	axrepndlem1 8979*	Lemma for the Axiom of Replacement with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)
|- ( -. A. y y = z -> E. x ( E. y A. z ( ph -> z = y ) -> A. z ( z e. x <-> E. x ( x e. y /\ A. y ph ) ) ) )
Theorem	axrepndlem2 8980	Lemma for the Axiom of Replacement with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
|- ( ( ( -. A. x x = y /\ -. A. x x = z ) /\ -. A. y y = z ) -> E. x ( E. y A. z ( ph -> z = y ) -> A. z ( z e. x <-> E. x ( x e. y /\ A. y ph ) ) ) )
Theorem	axrepnd 8981	A version of the Axiom of Replacement with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)
|- E. x ( E. y A. z ( ph -> z = y ) -> A. z ( A. y z e. x <-> E. x ( A. z x e. y /\ A. y ph ) ) )
Theorem	axunndlem1 8982*	Lemma for the Axiom of Union with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)
|- E. x A. y ( E. x ( y e. x /\ x e. z ) -> y e. x )
Theorem	axunnd 8983	A version of the Axiom of Union with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)
|- E. x A. y ( E. x ( y e. x /\ x e. z ) -> y e. x )
Theorem	axpowndlem1 8984	Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.)
|- ( A. x x = y -> ( -. x = y -> E. x A. y ( A. x ( E. z x e. y -> A. y x e. z ) -> y e. x ) ) )
Theorem	axpowndlem2 8985*	Lemma for the Axiom of Power Sets with no distinct variable conditions. Revised to remove a redundant antecedent from the consequence. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Mario Carneiro, 6-Dec-2016.) (Revised and shortened by Wolf Lammen, 9-Jun-2019.)
|- ( -. A. x x = y -> ( -. A. x x = z -> E. x A. y ( A. x ( E. z x e. y -> A. y x e. z ) -> y e. x ) ) )
Theorem	axpowndlem2OLD 8986*	Obsolete version of axpowndlem2 8985 as of 9-Jun-2019. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Mario Carneiro, 6-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( -. A. x x = y -> ( -. A. x x = z -> ( -. x = y -> E. x A. y ( A. x ( E. z x e. y -> A. y x e. z ) -> y e. x ) ) ) )
Theorem	axpowndlem3 8987*	Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.) (Revised by Mario Carneiro, 10-Dec-2016.) (Proof shortened by Wolf Lammen, 10-Jun-2019.)
|- ( -. x = y -> E. x A. y ( A. x ( E. z x e. y -> A. y x e. z ) -> y e. x ) )
Theorem	axpowndlem3OLD 8988*	Obsolete proof of axpowndlem3 8987 as of 9-Jun-2019. (Contributed by NM, 4-Jan-2002.) (Revised by Mario Carneiro, 10-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( -. x = y -> E. x A. y ( A. x ( E. z x e. y -> A. y x e. z ) -> y e. x ) )
Theorem	axpowndlem4 8989	Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)
|- ( -. A. y y = x -> ( -. A. y y = z -> ( -. x = y -> E. x A. y ( A. x ( E. z x e. y -> A. y x e. z ) -> y e. x ) ) ) )
Theorem	axpownd 8990	A version of the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.)
|- ( -. x = y -> E. x A. y ( A. x ( E. z x e. y -> A. y x e. z ) -> y e. x ) )
Theorem	axregndlem1 8991	Lemma for the Axiom of Regularity with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)
|- ( A. x x = z -> ( x e. y -> E. x ( x e. y /\ A. z ( z e. x -> -. z e. y ) ) ) )
Theorem	axregndlem2 8992*	Lemma for the Axiom of Regularity with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)
|- ( x e. y -> E. x ( x e. y /\ A. z ( z e. x -> -. z e. y ) ) )
Theorem	axregnd 8993	A version of the Axiom of Regularity with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.) (Proof shortened by Wolf Lammen, 18-Aug-2019.)
|- ( x e. y -> E. x ( x e. y /\ A. z ( z e. x -> -. z e. y ) ) )
Theorem	axregndOLD 8994	Obsolete proof of axregnd 8993 as of 18-Aug-2019. (Contributed by NM, 3-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( x e. y -> E. x ( x e. y /\ A. z ( z e. x -> -. z e. y ) ) )
Theorem	axinfndlem1 8995*	Lemma for the Axiom of Infinity with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 5-Jan-2002.)
|- ( A. x y e. z -> E. x ( y e. x /\ A. y ( y e. x -> E. z ( y e. z /\ z e. x ) ) ) )
Theorem	axinfnd 8996	A version of the Axiom of Infinity with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 5-Jan-2002.)
|- E. x ( y e. z -> ( y e. x /\ A. y ( y e. x -> E. z ( y e. z /\ z e. x ) ) ) )
Theorem	axacndlem1 8997	Lemma for the Axiom of Choice with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)
|- ( A. x x = y -> E. x A. y A. z ( A. x ( y e. z /\ z e. w ) -> E. w A. y ( E. w ( ( y e. z /\ z e. w ) /\ ( y e. w /\ w e. x ) ) <-> y = w ) ) )
Theorem	axacndlem2 8998	Lemma for the Axiom of Choice with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)
|- ( A. x x = z -> E. x A. y A. z ( A. x ( y e. z /\ z e. w ) -> E. w A. y ( E. w ( ( y e. z /\ z e. w ) /\ ( y e. w /\ w e. x ) ) <-> y = w ) ) )
Theorem	axacndlem3 8999	Lemma for the Axiom of Choice with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)
|- ( A. y y = z -> E. x A. y A. z ( A. x ( y e. z /\ z e. w ) -> E. w A. y ( E. w ( ( y e. z /\ z e. w ) /\ ( y e. w /\ w e. x ) ) <-> y = w ) ) )
Theorem	axacndlem4 9000*	Lemma for the Axiom of Choice with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 8-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)
|- E. x A. y A. z ( A. x ( y e. z /\ z e. w ) -> E. w A. y ( E. w ( ( y e. z /\ z e. w ) /\ ( y e. w /\ w e. x ) ) <-> y = w ) )