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	brdom6disj 8901*	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 8902	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 8903	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 8904	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 8905*	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 8906*	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 8907*	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 8908*	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 8909*	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 8910*	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 8911*	An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 8910 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 8912*	The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 8363 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 8913	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 8914	Any set is equinumerous to its cardinal number. Closed theorem form of cardid 8913. (Contributed by David Moews, 1-May-2017.)
|- ( A e. B -> ( card ` A ) ~~ A )
Theorem	cardidd 8915	Any set is equinumerous to its cardinal number. Deduction form of cardid 8913. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. B )   =>    |- ( ph -> ( card ` A ) ~~ A )
Theorem	cardf 8916	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 8917	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 12406 and the finite-set-only hashen 12405. 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 3323). 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 8304). (Contributed by NM, 22-Oct-2003.)
|- ( ( A e. C /\ B e. D ) -> ( ( card ` A ) = ( card ` B ) <-> A ~~ B ) )
Theorem	cardeq0 8918	Only the empty set has cardinality zero. (Contributed by NM, 23-Apr-2004.)
|- ( A e. V -> ( ( card ` A ) = (/) <-> A = (/) ) )
Theorem	unsnen 8919	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 8920	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 8921	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 8922	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 8923	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 8924	Trichotomy of dominance and strict dominance. (Contributed by NM, 4-Jan-2004.)
|- ( ( A e. V /\ B e. W ) -> ( A ~<_ B \/ B ~< A ) )
Theorem	entri3 8925	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 8926	A set strictly dominates iff its cardinal strictly dominates. (Contributed by NM, 30-Oct-2003.)
|- ( A ~< B <-> A ~< ( card ` B ) )
Theorem	canth3 8927	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 8928	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 8383. (Contributed by NM, 17-Sep-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)
|- ( om ~<_ A -> ( A X. A ) ~~ A )
Theorem	ondomon 8929*	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 7961. (Contributed by NM, 7-Nov-2003.) (Proof modification is discouraged.)
|- ( A e. V -> { x e. On | x ~<_ A } e. On )
Theorem	cardmin 8930*	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 8931	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 8932	Equivalence between two infiniteness criteria for sets. (Contributed by David Moews, 1-May-2017.)
|- ( A e. B -> ( -. A e. Fin <-> om ~<_ A ) )
Theorem	unirnfdomd 8933	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 8934*	Lemma for konigth 8935. (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 8935*	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 8936	The power set of an aleph dominates the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous, see gch3 9043 or gchaleph2 9039.) (Contributed by NM, 27-Aug-2005.)
|- ( aleph ` suc A ) ~<_ ~P ( aleph ` A )
Theorem	aleph1 8937	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 8938*	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 8939	A nonempty set that is a subset of its union is infinite. This version is proved from ax-ac 8830. See dominf 8816 for a version proved from ax-cc 8806. (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 8940*	The countable union of countable sets is countable (indexed union version of unictb 8941). (Contributed by Mario Carneiro, 18-Jan-2014.)
|- ( ( A ~<_ om /\ A. x e. A B ~<_ om ) -> U_ x e. A B ~<_ om )
Theorem	unictb 8941*	The countable union of countable sets is countable. Theorem 6Q of [Enderton] p. 159. See iunctb 8940 for indexed union version. (Contributed by NM, 26-Mar-2006.)
|- ( ( A ~<_ om /\ A. x e. A x ~<_ om ) -> U. A ~<_ om )
Theorem	infmap 8942*	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 8943	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 8944	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 8945	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 8946*	An alternate representation of a successor aleph. Compare alephsuc 8440 and alephsuc2 8452. Equality can be obtained by taking the card of the right-hand side then using alephcard 8442 and carden 8917. (Contributed by NM, 23-Oct-2004.)
|- ( A e. On -> ( aleph ` suc A ) ~~ { x e. On | x ~~ ( aleph ` A ) } )
Theorem	alephexp2 8947*	An expression equinumerous to 2 to an aleph power. The proof equates the two laws for cardinal exponentiation alephexp1 8945 (which works if the base is less than or equal to the exponent) and infmap 8942 (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 8948	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 8949*	A corollary of Konig's Theorem konigth 8935. 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 8950	A corollary of Konig's Theorem konigth 8935. 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 8951	From canth2 7663, 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 8935 (in the form of cfpwsdom 8950), ( 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 8952	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 8953	A lemma for proving conditionless ZFC axioms. (Contributed by NM, 1-Jan-2002.)
|- ( A. x x = y -> -. A. x y e. z )
Theorem	nd2 8954	A lemma for proving conditionless ZFC axioms. (Contributed by NM, 1-Jan-2002.)
|- ( A. x x = y -> -. A. x z e. y )
Theorem	nd3 8955	A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.)
|- ( A. x x = y -> -. A. z x e. y )
Theorem	nd4 8956	A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.)
|- ( A. x x = y -> -. A. z y e. x )
Theorem	axextnd 8957	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 8958*	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 8959	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 8960	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 8961*	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 8962	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 8963	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 8964*	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	axpowndlem3 8965*	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	axpowndlem4 8966	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 8967	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 8968	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 8969*	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 8970	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 8971	Obsolete proof of axregnd 8970 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 8972*	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 8973	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 8974	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 8975	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 8976	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 8977*	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 ) )
Theorem	axacndlem5 8978*	Lemma for the Axiom of Choice with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 3-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 ) )
Theorem	axacnd 8979	A version of the Axiom of Choice with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 3-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 ) )
Theorem	zfcndext 8980*	Axiom of Extensionality ax-ext 2432, reproved from conditionless ZFC version and predicate calculus. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)
|- ( A. z ( z e. x <-> z e. y ) -> x = y )
Theorem	zfcndrep 8981*	Axiom of Replacement ax-rep 4550, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)
|- ( A. w E. y A. z ( A. y ph -> z = y ) -> E. y A. z ( z e. y <-> E. w ( w e. x /\ A. y ph ) ) )
Theorem	zfcndun 8982*	Axiom of Union ax-un 6565, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)
|- E. y A. z ( E. w ( z e. w /\ w e. x ) -> z e. y )
Theorem	zfcndpow 8983*	Axiom of Power Sets ax-pow 4615, reproved from conditionless ZFC axioms. The proof uses the "Axiom of Twoness," dtru 4628. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)
|- E. y A. z ( A. w ( w e. z -> w e. x ) -> z e. y )
Theorem	zfcndreg 8984*	Axiom of Regularity ax-reg 8010, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)
|- ( E. y y e. x -> E. y ( y e. x /\ A. z ( z e. y -> -. z e. x ) ) )
Theorem	zfcndinf 8985*	Axiom of Infinity ax-inf 8046, reproved from conditionless ZFC axioms. Since we have already reproved Extensionality, Replacement, and Power Sets above, we are justified in referencing theorem el 4619 in the proof. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by NM, 15-Aug-2003.)
|- E. y ( x e. y /\ A. z ( z e. y -> E. w ( z e. w /\ w e. y ) ) )
Theorem	zfcndac 8986*	Axiom of Choice ax-ac 8830, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (New usage is discouraged.) (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 ) )
3.4  The Generalized Continuum Hypothesis
3.4.1  Sets satisfying the Generalized Continuum Hypothesis
Syntax	cgch 8987	Extend class notation to include the collection of sets that satisfy the GCH.
class GCH
Definition	df-gch 8988*	Define the collection of "GCH-sets", or sets for which the generalized continuum hypothesis holds. In this language the generalized continuum hypothesis can be expressed as GCH = _V. A set x satisfies the generalized continuum hypothesis if it is finite or there is no set y strictly between x and its powerset in cardinality. The continuum hypothesis is equivalent to om e. GCH. (Contributed by Mario Carneiro, 15-May-2015.)
|- GCH = ( Fin u. { x | A. y -. ( x ~< y /\ y ~< ~P x ) } )
Theorem	elgch 8989*	Elementhood in the collection of GCH-sets. (Contributed by Mario Carneiro, 15-May-2015.)
|- ( A e. V -> ( A e. GCH <-> ( A e. Fin \/ A. x -. ( A ~< x /\ x ~< ~P A ) ) ) )
Theorem	fingch 8990	A finite set is a GCH-set. (Contributed by Mario Carneiro, 15-May-2015.)
|- Fin C_ GCH
Theorem	gchi 8991	The only GCH-sets which have other sets between it and its power set are finite sets. (Contributed by Mario Carneiro, 15-May-2015.)
|- ( ( A e. GCH /\ A ~< B /\ B ~< ~P A ) -> A e. Fin )
Theorem	gchen1 8992	If A <_ B < ~P A, and A is an infinite GCH-set, then A = B in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)
|- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ B /\ B ~< ~P A ) ) -> A ~~ B )
Theorem	gchen2 8993	If A < B <_ ~P A, and A is an infinite GCH-set, then B = ~P A in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)
|- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~< B /\ B ~<_ ~P A ) ) -> B ~~ ~P A )
Theorem	gchor 8994	If A <_ B <_ ~P A, and A is an infinite GCH-set, then either A = B or B = ~P A in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)
|- ( ( ( A e. GCH /\ -. A e. Fin ) /\ ( A ~<_ B /\ B ~<_ ~P A ) ) -> ( A ~~ B \/ B ~~ ~P A ) )
Theorem	engch 8995	The property of being a GCH-set is a cardinal invariant. (Contributed by Mario Carneiro, 15-May-2015.)
|- ( A ~~ B -> ( A e. GCH <-> B e. GCH ) )
Theorem	gchdomtri 8996	Under certain conditions, a GCH-set can demonstrate trichotomy of dominance. Lemma for gchac 9048. (Contributed by Mario Carneiro, 15-May-2015.)
|- ( ( A e. GCH /\ ( A +c A ) ~~ A /\ B ~<_ ~P A ) -> ( A ~<_ B \/ B ~<_ A ) )
Theorem	fpwwe2cbv 8997*	Lemma for fpwwe2 9010. (Contributed by Mario Carneiro, 3-Jun-2015.)
|- W = { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x [. ( `' r " { y } ) / u ]. ( u F ( r i^i ( u X. u ) ) ) = y ) ) }   =>    |- W = { <. a , s >. | ( ( a C_ A /\ s C_ ( a X. a ) ) /\ ( s We a /\ A. z e. a [. ( `' s " { z } ) / v ]. ( v F ( s i^i ( v X. v ) ) ) = z ) ) }
Theorem	fpwwe2lem1 8998*	Lemma for fpwwe2 9010. (Contributed by Mario Carneiro, 15-May-2015.)
|- W = { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x [. ( `' r " { y } ) / u ]. ( u F ( r i^i ( u X. u ) ) ) = y ) ) }   =>    |- W C_ ( ~P A X. ~P ( A X. A ) )
Theorem	fpwwe2lem2 8999*	Lemma for fpwwe2 9010. (Contributed by Mario Carneiro, 19-May-2015.)
|- W = { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x [. ( `' r " { y } ) / u ]. ( u F ( r i^i ( u X. u ) ) ) = y ) ) }   &    |- ( ph -> A e. _V )   =>    |- ( ph -> ( X W R <-> ( ( X C_ A /\ R C_ ( X X. X ) ) /\ ( R We X /\ A. y e. X [. ( `' R " { y } ) / u ]. ( u F ( R i^i ( u X. u ) ) ) = y ) ) ) )
Theorem	fpwwe2lem3 9000*	Lemma for fpwwe2 9010. (Contributed by Mario Carneiro, 19-May-2015.)
|- W = { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x [. ( `' r " { y } ) / u ]. ( u F ( r i^i ( u X. u ) ) ) = y ) ) }   &    |- ( ph -> A e. _V )   &    |- ( ph -> X W R )   =>    |- ( ( ph /\ B e. X ) -> ( ( `' R " { B } ) F ( R i^i ( ( `' R " { B } ) X. ( `' R " { B } ) ) ) ) = B )