P. 107 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 10601-10700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fodom 10601	An onto function implies dominance of domain over range. (Contributed by NM, 23-Jul-2004.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐹:𝐴–onto→𝐵 → 𝐵 ≼ 𝐴)
Theorem	dmct 10602	The domain of a countable set is countable. The proof uses fodomnum 10136 rather than fodomg 10600, and so does not require ax-ac 10537. (Contributed by Thierry Arnoux, 29-Dec-2016.) (Revised by Vincent Gonzalez, 24-Aug-2026.)
⊢ (𝐴 ≼ ω → dom 𝐴 ≼ ω)
Theorem	dmctOLD 10603	Obsolete version of dmct 10602 as of 26-Aug-2026. (Contributed by Thierry Arnoux, 29-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ≼ ω → dom 𝐴 ≼ ω)
Theorem	rnct 10604	The range of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
⊢ (𝐴 ≼ ω → ran 𝐴 ≼ ω)
Theorem	fodomb 10605*	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 ≠ ∅ ∧ ∃𝑓 𝑓:𝐴–onto→𝐵) ↔ (∅ ≺ 𝐵 ∧ 𝐵 ≼ 𝐴))
Theorem	wdomac 10606	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 10607*	Equivalence to a dominance relation. (Contributed by NM, 27-Mar-2007.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓(∀𝑥∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥))
Theorem	brdom5 10608*	An equivalence to a dominance relation. (Contributed by NM, 29-Mar-2007.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥))
Theorem	brdom4 10609*	An equivalence to a dominance relation. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥))
Theorem	brdom7disj 10610*	An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 29-Mar-2007.) (Revised by NM, 16-Jun-2017.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ (𝐴 ∩ 𝐵) = ∅    ⇒   ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓))
Theorem	brdom6disj 10611*	An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 5-Apr-2007.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ (𝐴 ∩ 𝐵) = ∅    ⇒   ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓))
Theorem	fin71ac 10612	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 10613	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 ∈ 𝐵 → (Fun 𝐹 → (𝐹 “ 𝐴) ≼ 𝐴))
Theorem	imadomnum 10614	A version of imadomg 10613 that does not require the axiom of choice ax-ac 10537. (Contributed by Vincent Gonzalez, 25-Aug-2026.)
⊢ (𝐴 ∈ dom card → (Fun 𝐹 → (𝐹 “ 𝐴) ≼ 𝐴))
Theorem	fimact 10615	The image by a function of a countable set is countable. The proof uses imadomnum 10614 rather than imadomg 10613, and so does not require ax-ac 10537. (Contributed by Thierry Arnoux, 27-Mar-2018.) (Revised by Vincent Gonzalez, 25-Aug-2026.)
⊢ ((𝐴 ≼ ω ∧ Fun 𝐹) → (𝐹 “ 𝐴) ≼ ω)
Theorem	fimactOLD 10616	Obsolete version of fimact 10615 as of 26-Aug-2026. (Contributed by Thierry Arnoux, 27-Mar-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ≼ ω ∧ Fun 𝐹) → (𝐹 “ 𝐴) ≼ ω)
Theorem	fnrndomnum 10617	A version of fnrndomg 10618 that does not require the axiom of choice ax-ac 10537. (Contributed by Vincent Gonzalez, 17-Aug-2026.)
⊢ (𝐴 ∈ dom card → (𝐹 Fn 𝐴 → ran 𝐹 ≼ 𝐴))
Theorem	fnrndomg 10618	The range of a function is dominated by its domain. This theorem requires the axiom of choice ax-ac2 10541; see fnrndomnum 10617 for a version that does not. (Contributed by NM, 1-Sep-2004.) (Proof shortened by Vincent Gonzalez, 17-Aug-2026.)
⊢ (𝐴 ∈ 𝐵 → (𝐹 Fn 𝐴 → ran 𝐹 ≼ 𝐴))
Theorem	fnrndomgOLD 10619	Obsolete version of fnrndomg 10618 as of 18-Aug-2026. (Contributed by NM, 1-Sep-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝐵 → (𝐹 Fn 𝐴 → ran 𝐹 ≼ 𝐴))
Theorem	fnct 10620	If the domain of a function is countable, the function is countable. The proof uses fnrndomnum 10617 rather than fnrndomg 10618, and so does not require ax-ac 10537. (Contributed by Thierry Arnoux, 29-Dec-2016.) (Revised by Vincent Gonzalez, 24-Aug-2026.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐹 ≼ ω)
Theorem	fnctOLD 10621	Obsolete version of fnct 10620 as of 26-Aug-2026. (Contributed by Thierry Arnoux, 29-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐹 ≼ ω)
Theorem	mptct 10622*	A countable mapping set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
⊢ (𝐴 ≼ ω → (𝑥 ∈ 𝐴 ↦ 𝐵) ≼ ω)
Theorem	iunfo 10623*	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 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)    ⇒   ⊢ (2nd ↾ 𝑇):𝑇–onto→∪ 𝑥 ∈ 𝐴 𝐵
Theorem	iundom2g 10624*	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 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)    &   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵) ∈ AC 𝐴)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ≼ 𝐶)    ⇒   ⊢ (𝜑 → 𝑇 ≼ (𝐴 × 𝐶))
Theorem	iundomg 10625*	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 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)    &   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵) ∈ AC 𝐴)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ≼ 𝐶)    &   ⊢ (𝜑 → (𝐴 × 𝐶) ∈ AC ∪ 𝑥 ∈ 𝐴 𝐵)    ⇒   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ≼ (𝐴 × 𝐶))
Theorem	iundom 10626*	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 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ≼ 𝐵) → ∪ 𝑥 ∈ 𝐴 𝐶 ≼ (𝐴 × 𝐵))
Theorem	unidom 10627*	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 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ 𝐵) → ∪ 𝐴 ≼ (𝐴 × 𝐵))
Theorem	uniimadom 10628*	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 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((Fun 𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ≼ 𝐵) → ∪ (𝐹 “ 𝐴) ≼ (𝐴 × 𝐵))
Theorem	uniimadomf 10629*	An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 10628 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by NM, 26-Mar-2006.)
⊢ Ⅎ𝑥𝐹    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((Fun 𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ≼ 𝐵) → ∪ (𝐹 “ 𝐴) ≼ (𝐴 × 𝐵))
3.2.3  Cardinal number theorems using Axiom of Choice
Theorem	cardval 10630*	The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 10072 for a simpler version of its value. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴}
Theorem	cardid 10631	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 ∈ V    ⇒   ⊢ (card‘𝐴) ≈ 𝐴
Theorem	cardidg 10632	Any set is equinumerous to its cardinal number. Closed theorem form of cardid 10631. (Contributed by David Moews, 1-May-2017.)
⊢ (𝐴 ∈ 𝐵 → (card‘𝐴) ≈ 𝐴)
Theorem	cardidd 10633	Any set is equinumerous to its cardinal number. Deduction form of cardid 10631. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (card‘𝐴) ≈ 𝐴)
Theorem	cardf 10634	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 10635	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 14492 and the finite-set-only hashen 14491. 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 3737). 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 3737. 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 9959). (Contributed by NM, 22-Oct-2003.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((card‘𝐴) = (card‘𝐵) ↔ 𝐴 ≈ 𝐵))
Theorem	cardeq0 10636	Only the empty set has cardinality zero. (Contributed by NM, 23-Apr-2004.)
⊢ (𝐴 ∈ 𝑉 → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
Theorem	unsnen 10637	Equinumerosity of a set with a new element added. (Contributed by NM, 7-Nov-2008.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐴 ∪ {𝐵}) ≈ suc (card‘𝐴))
Theorem	carddom 10638	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 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))
Theorem	cardsdom 10639	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 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((card‘𝐴) ∈ (card‘𝐵) ↔ 𝐴 ≺ 𝐵))
Theorem	domtri 10640	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 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴))
Theorem	entric 10641	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 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵 ∨ 𝐵 ≺ 𝐴))
Theorem	entri2 10642	Trichotomy of dominance and strict dominance. (Contributed by NM, 4-Jan-2004.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≼ 𝐵 ∨ 𝐵 ≺ 𝐴))
Theorem	entri3 10643	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 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≼ 𝐵 ∨ 𝐵 ≼ 𝐴))
Theorem	sdomsdomcard 10644	A set strictly dominates iff its cardinal strictly dominates. (Contributed by NM, 30-Oct-2003.)
⊢ (𝐴 ≺ 𝐵 ↔ 𝐴 ≺ (card‘𝐵))
Theorem	canth3 10645	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 ∈ 𝑉 → (card‘𝐴) ∈ (card‘𝒫 𝐴))
Theorem	infxpidm 10646	Every infinite class is equinumerous to its Cartesian square. This theorem, which is equivalent to the axiom of choice over ZF, provides the basis for infinite cardinal arithmetic. Proposition 10.40 of [TakeutiZaring] p. 95. This is a corollary of infxpen 10093 (used via infxpidm2 10096). (Contributed by NM, 17-Sep-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)
⊢ (ω ≼ 𝐴 → (𝐴 × 𝐴) ≈ 𝐴)
Theorem	ondomon 10647*	The class of ordinals dominated by a given set is an ordinal. Theorem 56 of [Suppes] p. 227. This theorem can be proved without the axiom of choice, see hartogs 9538. (Contributed by NM, 7-Nov-2003.) (Proof modification is discouraged.) Use hartogs 9538 instead. (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ On)
Theorem	cardmin 10648*	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 ∈ 𝑉 → (card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
Theorem	ficard 10649	A set is finite iff its cardinal is a natural number. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin ↔ (card‘𝐴) ∈ ω))
Theorem	infinfg 10650	Equivalence between two infiniteness criteria for sets. To avoid the axiom of infinity, we include it as a hypothesis. (Contributed by Scott Fenton, 20-Feb-2026.)
⊢ ((ω ∈ V ∧ 𝐴 ∈ 𝐵) → (¬ 𝐴 ∈ Fin ↔ ω ≼ 𝐴))
Theorem	infinf 10651	Equivalence between two infiniteness criteria for sets. (Contributed by David Moews, 1-May-2017.) (Proof shortened by Scott Fenton, 20-Feb-2026.)
⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ Fin ↔ ω ≼ 𝐴))
Theorem	unirnfdomd 10652	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.)
⊢ (𝜑 → 𝐹:𝑇⟶Fin)    &   ⊢ (𝜑 → ¬ 𝑇 ∈ Fin)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∪ ran 𝐹 ≼ 𝑇)
Theorem	konigthlem 10653*	Lemma for konigth 10654. (Contributed by Mario Carneiro, 22-Feb-2013.)
⊢ 𝐴 ∈ V    &   ⊢ 𝑆 = ∪ 𝑖 ∈ 𝐴 (𝑀‘𝑖)    &   ⊢ 𝑃 = X𝑖 ∈ 𝐴 (𝑁‘𝑖)    &   ⊢ 𝐷 = (𝑖 ∈ 𝐴 ↦ (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖)))    &   ⊢ 𝐸 = (𝑖 ∈ 𝐴 ↦ (𝑒‘𝑖))    ⇒   ⊢ (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) ≺ (𝑁‘𝑖) → 𝑆 ≺ 𝑃)
Theorem	konigth 10654*	Konig's Theorem. If 𝑚(𝑖) ≺ 𝑛(𝑖) for all 𝑖 ∈ 𝐴, then Σ𝑖 ∈ 𝐴𝑚(𝑖) ≺ ∏𝑖 ∈ 𝐴𝑛(𝑖), where the sums and products stand in for disjoint union and infinite cartesian product. The version here is proven with 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 ∈ V    &   ⊢ 𝑆 = ∪ 𝑖 ∈ 𝐴 (𝑀‘𝑖)    &   ⊢ 𝑃 = X𝑖 ∈ 𝐴 (𝑁‘𝑖)    ⇒   ⊢ (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) ≺ (𝑁‘𝑖) → 𝑆 ≺ 𝑃)
Theorem	alephsucpw 10655	The power set of an aleph dominates the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous, see gch3 10761 or gchaleph2 10757.) (Contributed by NM, 27-Aug-2005.)
⊢ (ℵ‘suc 𝐴) ≼ 𝒫 (ℵ‘𝐴)
Theorem	aleph1 10656	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.)
⊢ (ℵ‘1o) ≼ (2o ↑m (ℵ‘∅))
Theorem	alephval2 10657*	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 ∈ On ∧ ∅ ∈ 𝐴) → (ℵ‘𝐴) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥})
Theorem	dominfac 10658	A nonempty set that is a subset of its union is infinite. This version is proved from ax-ac 10537. See dominf 10523 for a version proved from ax-cc 10513. (Contributed by NM, 25-Mar-2007.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ ∪ 𝐴) → ω ≼ 𝐴)
3.2.4  Cardinal number arithmetic using Axiom of Choice
Theorem	iunctb 10659*	The countable union of countable sets is countable (indexed union version of unictb 10660). (Contributed by Mario Carneiro, 18-Jan-2014.)
⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ ω) → ∪ 𝑥 ∈ 𝐴 𝐵 ≼ ω)
Theorem	unictb 10660*	The countable union of countable sets is countable. Theorem 6Q of [Enderton] p. 159. See iunctb 10659 for indexed union version. (Contributed by NM, 26-Mar-2006.)
⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ω) → ∪ 𝐴 ≼ ω)
Theorem	infmap 10661*	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.)
⊢ ((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴) → (𝐴 ↑m 𝐵) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)})
Theorem	alephadd 10662	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.)
⊢ ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵))
Theorem	alephmul 10663	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 ∈ On ∧ 𝐵 ∈ On) → ((ℵ‘𝐴) × (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))
Theorem	alephexp1 10664	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 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ((ℵ‘𝐴) ↑m (ℵ‘𝐵)) ≈ (2o ↑m (ℵ‘𝐵)))
Theorem	alephsuc3 10665*	An alternate representation of a successor aleph. Compare alephsuc 10147 and alephsuc2 10159. Equality can be obtained by taking the card of the right-hand side then using alephcard 10149 and carden 10635. (Contributed by NM, 23-Oct-2004.)
⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) ≈ {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)})
Theorem	alephexp2 10666*	An expression equinumerous to 2 to an aleph power. The proof equates the two laws for cardinal exponentiation alephexp1 10664 (which works if the base is less than or equal to the exponent) and infmap 10661 (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 ∈ On → (2o ↑m (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))})
3.2.5  Cofinality using the Axiom of Choice
Theorem	alephreg 10667	A successor aleph is regular. Theorem 11.15 of [TakeutiZaring] p. 103. (Contributed by Mario Carneiro, 9-Mar-2013.)
⊢ (cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)
Theorem	pwcfsdom 10668*	A corollary of Konig's Theorem konigth 10654. Theorem 11.28 of [TakeutiZaring] p. 108. (Contributed by Mario Carneiro, 20-Mar-2013.)
⊢ 𝐻 = (𝑦 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓‘𝑦)))    ⇒   ⊢ (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑m (cf‘(ℵ‘𝐴)))
Theorem	cfpwsdom 10669	A corollary of Konig's Theorem konigth 10654. Theorem 11.29 of [TakeutiZaring] p. 108. (Contributed by Mario Carneiro, 20-Mar-2013.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (2o ≼ 𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))))
Theorem	alephom 10670	From canth2 9149, we know that (ℵ‘0) < (2↑ω), but we cannot prove that (2↑ω) = (ℵ‘1) (this is the Continuum Hypothesis), nor can we prove that it is less than any bound whatsoever (i.e. the statement (ℵ‘𝐴) < (2↑ω) is consistent for any ordinal 𝐴). However, we can prove that (2↑ω) is not equal to (ℵ‘ω), nor (ℵ‘(ℵ‘ω)), on cofinality grounds, because by Konig's Theorem konigth 10654 (in the form of cfpwsdom 10669), (2↑ω) has uncountable cofinality, which eliminates limit alephs like (ℵ‘ω). (The first limit aleph that is not eliminated is (ℵ‘(ℵ‘1)), which has cofinality (ℵ‘1).) (Contributed by Mario Carneiro, 21-Mar-2013.)
⊢ (card‘(2o ↑m ω)) ≠ (ℵ‘ω)
Theorem	smobeth 10671	The beth function is strictly monotone. This function is not strictly the beth function, but rather bethA is the same as (card‘(𝑅1‘(ω +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 ∘ 𝑅1)
3.3  ZFC Axioms with no distinct variable requirements
Theorem	nd1 10672	A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2401. (Contributed by NM, 1-Jan-2002.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑦 ∈ 𝑧)
Theorem	nd2 10673	A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2401. (Contributed by NM, 1-Jan-2002.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑧 ∈ 𝑦)
Theorem	nd3 10674	A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.)
⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑥 ∈ 𝑦)
Theorem	nd4 10675	A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2401. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑦 ∈ 𝑥)
Theorem	axextnd 10676	A version of the Axiom of Extensionality with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2401. (Contributed by NM, 14-Aug-2003.) (New usage is discouraged.)
⊢ ∃𝑥((𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) → 𝑦 = 𝑧)
Theorem	axrepndlem1 10677*	Lemma for the Axiom of Replacement with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2401. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.)
⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))))
Theorem	axrepndlem2 10678	Lemma for the Axiom of Replacement with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2401. (Contributed by NM, 2-Jan-2002.) (Proof shortened by Mario Carneiro, 6-Dec-2016.) (New usage is discouraged.)
⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))))
Theorem	axrepnd 10679	A version of the Axiom of Replacement with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2401. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.)
⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧 ∈ 𝑥 ↔ ∃𝑥(∀𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))
Theorem	axunndlem1 10680*	Lemma for the Axiom of Union with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2401. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.)
⊢ ∃𝑥∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)
Theorem	axunnd 10681	A version of the Axiom of Union with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2401. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.)
⊢ ∃𝑥∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)
Theorem	axpowndlem1 10682	Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.)
⊢ (∀𝑥 𝑥 = 𝑦 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
Theorem	axpowndlem2 10683*	Lemma for the Axiom of Power Sets with no distinct variable conditions. Revised to remove a redundant antecedent from the consequence. Usage of this theorem is discouraged because it depends on ax-13 2401. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Mario Carneiro, 6-Dec-2016.) (Revised and shortened by Wolf Lammen, 9-Jun-2019.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))
Theorem	axpowndlem3 10684*	Lemma for the Axiom of Power Sets with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2401. (Contributed by NM, 4-Jan-2002.) (Revised by Mario Carneiro, 10-Dec-2016.) (Proof shortened by Wolf Lammen, 10-Jun-2019.) (New usage is discouraged.)
⊢ (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
Theorem	axpowndlem4 10685	Lemma for the Axiom of Power Sets with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2401. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.) (New usage is discouraged.)
⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))))
Theorem	axpownd 10686	A version of the Axiom of Power Sets with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2401. (Contributed by NM, 4-Jan-2002.) (New usage is discouraged.)
⊢ (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
Theorem	axregndlem1 10687	Lemma for the Axiom of Regularity with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2401. (Contributed by NM, 3-Jan-2002.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑧 → (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))))
Theorem	axregndlem2 10688*	Lemma for the Axiom of Regularity with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2401. (Contributed by NM, 3-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.) (New usage is discouraged.)
⊢ (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))
Theorem	axregnd 10689	A version of the Axiom of Regularity with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2401. (Contributed by NM, 3-Jan-2002.) (Proof shortened by Wolf Lammen, 18-Aug-2019.) (New usage is discouraged.)
⊢ (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))
Theorem	axinfndlem1 10690*	Lemma for the Axiom of Infinity with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 5-Jan-2002.)
⊢ (∀𝑥 𝑦 ∈ 𝑧 → ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))))
Theorem	axinfnd 10691	A version of the Axiom of Infinity with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 5-Jan-2002.)
⊢ ∃𝑥(𝑦 ∈ 𝑧 → (𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))))
Theorem	axacndlem1 10692	Lemma for the Axiom of Choice with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2401. (Contributed by NM, 3-Jan-2002.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∃𝑥∀𝑦∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤)))
Theorem	axacndlem2 10693	Lemma for the Axiom of Choice with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2401. (Contributed by NM, 3-Jan-2002.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑧 → ∃𝑥∀𝑦∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤)))
Theorem	axacndlem3 10694	Lemma for the Axiom of Choice with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2401. (Contributed by NM, 3-Jan-2002.) (New usage is discouraged.)
⊢ (∀𝑦 𝑦 = 𝑧 → ∃𝑥∀𝑦∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤)))
Theorem	axacndlem4 10695*	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.)
⊢ ∃𝑥∀𝑦∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤))
Theorem	axacndlem5 10696*	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.)
⊢ ∃𝑥∀𝑦∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤))
Theorem	axacnd 10697	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.)
⊢ ∃𝑥∀𝑦∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤))
Theorem	zfcndext 10698*	Axiom of Extensionality ax-ext 2732, reproved from conditionless ZFC version and predicate calculus. Usage of this theorem is discouraged because it depends on ax-13 2401. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
Theorem	zfcndrep 10699*	Axiom of Replacement ax-rep 5231, reproved from conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2401. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
Theorem	zfcndun 10700*	Axiom of Union ax-un 7751, reproved from conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2401. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)