P. 101 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 10001-10100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	onenon 10001	Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
⊢ (𝐴 ∈ On → 𝐴 ∈ dom card)
Theorem	tskwe 10002*	A Tarski set is well-orderable. (Contributed by Mario Carneiro, 19-Apr-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴) → 𝐴 ∈ dom card)
Theorem	xpnum 10003	The cartesian product of numerable sets is numerable. (Contributed by Mario Carneiro, 3-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 × 𝐵) ∈ dom card)
Theorem	cardval3 10004*	An alternate definition of the value of (card‘𝐴) that does not require AC to prove. (Contributed by Mario Carneiro, 7-Jan-2013.) (Revised by Mario Carneiro, 27-Apr-2015.)
⊢ (𝐴 ∈ dom card → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴})
Theorem	cardid2 10005	Any numerable set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)
⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
Theorem	isnum3 10006	A set is numerable iff it is equinumerous with its cardinal. (Contributed by Mario Carneiro, 29-Apr-2015.)
⊢ (𝐴 ∈ dom card ↔ (card‘𝐴) ≈ 𝐴)
Theorem	oncardval 10007*	The value of the cardinal number function with an ordinal number as its argument. Unlike cardval 10601, this theorem does not require the Axiom of Choice. (Contributed by NM, 24-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)
⊢ (𝐴 ∈ On → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴})
Theorem	oncardid 10008	Any ordinal number is equinumerous to its cardinal number. Unlike cardid 10602, this theorem does not require the Axiom of Choice. (Contributed by NM, 26-Jul-2004.)
⊢ (𝐴 ∈ On → (card‘𝐴) ≈ 𝐴)
Theorem	cardonle 10009	The cardinal of an ordinal number is less than or equal to the ordinal number. Proposition 10.6(3) of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.)
⊢ (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
Theorem	card0 10010	The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.)
⊢ (card‘∅) = ∅
Theorem	cardidm 10011	The cardinality function is idempotent. Proposition 10.11 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)
⊢ (card‘(card‘𝐴)) = (card‘𝐴)
Theorem	oncard 10012*	A set is a cardinal number iff it equals its own cardinal number. Proposition 10.9 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)
⊢ (∃𝑥 𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴))
Theorem	ficardom 10013	The cardinal number of a finite set is a finite ordinal. (Contributed by Paul Chapman, 11-Apr-2009.) (Revised by Mario Carneiro, 4-Feb-2013.)
⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
Theorem	ficardid 10014	A finite set is equinumerous to its cardinal number. (Contributed by Mario Carneiro, 21-Sep-2013.)
⊢ (𝐴 ∈ Fin → (card‘𝐴) ≈ 𝐴)
Theorem	cardnn 10015	The cardinality of a natural number is the number. Corollary 10.23 of [TakeutiZaring] p. 90. (Contributed by Mario Carneiro, 7-Jan-2013.)
⊢ (𝐴 ∈ ω → (card‘𝐴) = 𝐴)
Theorem	cardnueq0 10016	The empty set is the only numerable set with cardinality zero. (Contributed by Mario Carneiro, 7-Jan-2013.)
⊢ (𝐴 ∈ dom card → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
Theorem	cardne 10017	No member of a cardinal number of a set is equinumerous to the set. Proposition 10.6(2) of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 9-Jan-2013.)
⊢ (𝐴 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵)
Theorem	carden2a 10018	If two sets have equal nonzero cardinalities, then they are equinumerous. This assertion and carden2b 10019 are meant to replace carden 10606 in ZF without AC. (Contributed by Mario Carneiro, 9-Jan-2013.)
⊢ (((card‘𝐴) = (card‘𝐵) ∧ (card‘𝐴) ≠ ∅) → 𝐴 ≈ 𝐵)
Theorem	carden2b 10019	If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 10018 are meant to replace carden 10606 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
⊢ (𝐴 ≈ 𝐵 → (card‘𝐴) = (card‘𝐵))
Theorem	card1 10020*	A set has cardinality one iff it is a singleton. (Contributed by Mario Carneiro, 10-Jan-2013.)
⊢ ((card‘𝐴) = 1o ↔ ∃𝑥 𝐴 = {𝑥})
Theorem	cardsn 10021	A singleton has cardinality one. (Contributed by Mario Carneiro, 10-Jan-2013.)
⊢ (𝐴 ∈ 𝑉 → (card‘{𝐴}) = 1o)
Theorem	carddomi2 10022	Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 10609, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴 ≼ 𝐵))
Theorem	sdomsdomcardi 10023	A set strictly dominates if its cardinal strictly dominates. (Contributed by Mario Carneiro, 13-Jan-2013.)
⊢ (𝐴 ≺ (card‘𝐵) → 𝐴 ≺ 𝐵)
Theorem	cardlim 10024	An infinite cardinal is a limit ordinal. Equivalent to Exercise 4 of [TakeutiZaring] p. 91. (Contributed by Mario Carneiro, 13-Jan-2013.)
⊢ (ω ⊆ (card‘𝐴) ↔ Lim (card‘𝐴))
Theorem	cardsdomelir 10025	A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 10026 and can be proven without the AC. (Contributed by Mario Carneiro, 15-Jan-2013.)
⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ≺ 𝐵)
Theorem	cardsdomel 10026	A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 15-Jan-2013.) (Revised by Mario Carneiro, 4-Jun-2015.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ dom card) → (𝐴 ≺ 𝐵 ↔ 𝐴 ∈ (card‘𝐵)))
Theorem	iscard 10027*	Two ways to express the property of being a cardinal number. (Contributed by Mario Carneiro, 15-Jan-2013.)
⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ 𝐴 𝑥 ≺ 𝐴))
Theorem	iscard2 10028*	Two ways to express the property of being a cardinal number. Definition 8 of [Suppes] p. 225. (Contributed by Mario Carneiro, 15-Jan-2013.)
⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ On (𝐴 ≈ 𝑥 → 𝐴 ⊆ 𝑥)))
Theorem	carddom2 10029	Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 10609, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))
Theorem	harcard 10030	The class of ordinal numbers dominated by a set is a cardinal number. Theorem 59 of [Suppes] p. 228. (Contributed by Mario Carneiro, 20-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)
⊢ (card‘(har‘𝐴)) = (har‘𝐴)
Theorem	cardprclem 10031*	Lemma for cardprc 10032. (Contributed by Mario Carneiro, 22-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)
⊢ 𝐴 = {𝑥 ∣ (card‘𝑥) = 𝑥}    ⇒   ⊢ ¬ 𝐴 ∈ V
Theorem	cardprc 10032	The class of all cardinal numbers is not a set (i.e. is a proper class). Theorem 19.8 of [Eisenberg] p. 310. In this proof (which does not use AC), we cannot use Cantor's construction canth3 10616 to ensure that there is always a cardinal larger than a given cardinal, but we can use Hartogs' construction hartogs 9516 to construct (effectively) (ℵ‘suc 𝐴) from (ℵ‘𝐴), which achieves the same thing. (Contributed by Mario Carneiro, 22-Jan-2013.)
⊢ {𝑥 ∣ (card‘𝑥) = 𝑥} ∉ V
Theorem	carduni 10033*	The union of a set of cardinals is a cardinal. Theorem 18.14 of [Monk1] p. 133. (Contributed by Mario Carneiro, 20-Jan-2013.)
⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥 → (card‘∪ 𝐴) = ∪ 𝐴))
Theorem	cardiun 10034*	The indexed union of a set of cardinals is a cardinal. (Contributed by NM, 3-Nov-2003.)
⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵 → (card‘∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 𝐵))
Theorem	cardennn 10035	If 𝐴 is equinumerous to a natural number, then that number is its cardinal. (Contributed by Mario Carneiro, 11-Jan-2013.)
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ ω) → (card‘𝐴) = 𝐵)
Theorem	cardsucinf 10036	The cardinality of the successor of an infinite ordinal. (Contributed by Mario Carneiro, 11-Jan-2013.)
⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (card‘suc 𝐴) = (card‘𝐴))
Theorem	cardsucnn 10037	The cardinality of the successor of a finite ordinal (natural number). This theorem does not hold for infinite ordinals; see cardsucinf 10036. (Contributed by NM, 7-Nov-2008.)
⊢ (𝐴 ∈ ω → (card‘suc 𝐴) = suc (card‘𝐴))
Theorem	cardom 10038	The set of natural numbers is a cardinal number. Theorem 18.11 of [Monk1] p. 133. (Contributed by NM, 28-Oct-2003.)
⊢ (card‘ω) = ω
Theorem	carden2 10039	Two numerable sets are equinumerous iff their cardinal numbers are equal. Unlike carden 10606, the Axiom of Choice is not required. (Contributed by Mario Carneiro, 22-Sep-2013.)
⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) = (card‘𝐵) ↔ 𝐴 ≈ 𝐵))
Theorem	cardsdom2 10040	A numerable set is strictly dominated by another iff their cardinalities are strictly ordered. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ∈ (card‘𝐵) ↔ 𝐴 ≺ 𝐵))
Theorem	domtri2 10041	Trichotomy of dominance for numerable sets (does not use AC). (Contributed by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴))
Theorem	nnsdomel 10042	Strict dominance and elementhood are the same for finite ordinals. (Contributed by Stefan O'Rear, 2-Nov-2014.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ 𝐴 ≺ 𝐵))
Theorem	cardval2 10043*	An alternate version of the value of the cardinal number of a set. Compare cardval 10601. This theorem could be used to give a simpler definition of card in place of df-card 9991. It apparently does not occur in the literature. (Contributed by NM, 7-Nov-2003.)
⊢ (𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴})
Theorem	isinffi 10044*	An infinite set contains subsets equinumerous to every finite set. Extension of isinf 9234 from finite ordinals to all finite sets. (Contributed by Stefan O'Rear, 8-Oct-2014.)
⊢ ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐵–1-1→𝐴)
Theorem	fidomtri 10045	Trichotomy of dominance without AC when one set is finite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 27-Apr-2015.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) → (𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴))
Theorem	fidomtri2 10046	Trichotomy of dominance without AC when one set is finite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 7-May-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin) → (𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴))
Theorem	harsdom 10047	The Hartogs number of a well-orderable set strictly dominates the set. (Contributed by Mario Carneiro, 15-May-2015.)
⊢ (𝐴 ∈ dom card → 𝐴 ≺ (har‘𝐴))
Theorem	onsdom 10048*	Any well-orderable set is strictly dominated by an ordinal number. (Contributed by Jeff Hankins, 22-Oct-2009.) (Proof shortened by Mario Carneiro, 15-May-2015.)
⊢ (𝐴 ∈ dom card → ∃𝑥 ∈ On 𝐴 ≺ 𝑥)
Theorem	harval2 10049*	An alternate expression for the Hartogs number of a well-orderable set. (Contributed by Mario Carneiro, 15-May-2015.)
⊢ (𝐴 ∈ dom card → (har‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
Theorem	harsucnn 10050	The next cardinal after a finite ordinal is the successor ordinal. (Contributed by RP, 5-Nov-2023.)
⊢ (𝐴 ∈ ω → (har‘𝐴) = suc 𝐴)
Theorem	cardmin2 10051*	The smallest ordinal that strictly dominates a set is a cardinal, if it exists. (Contributed by Mario Carneiro, 2-Feb-2013.)
⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 ↔ (card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
Theorem	pm54.43lem 10052*	In Theorem *54.43 of [WhiteheadRussell] p. 360, the number 1 is defined as the collection of all sets with cardinality 1 (i.e. all singletons; see card1 10020), so that their 𝐴 ∈ 1 means, in our notation, 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o}. Here we show that this is equivalent to 𝐴 ≈ 1o so that we can use the latter more convenient notation in pm54.43 10053. (Contributed by NM, 4-Nov-2013.)
⊢ (𝐴 ≈ 1o ↔ 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o})
Theorem	pm54.43 10053	Theorem *54.43 of [WhiteheadRussell] p. 360. "From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2." See http://en.wikipedia.org/wiki/Principia_Mathematica#Quotations. This theorem states that two sets of cardinality 1 are disjoint iff their union has cardinality 2. Whitehead and Russell define 1 as the collection of all sets with cardinality 1 (i.e. all singletons; see card1 10020), so that their 𝐴 ∈ 1 means, in our notation, 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o} which is the same as 𝐴 ≈ 1o by pm54.43lem 10052. We do not have several of their earlier lemmas available (which would otherwise be unused by our different approach to arithmetic), so our proof is longer. (It is also longer because we must show every detail.) Theorem dju1p1e2 10223 shows the derivation of 1+1=2 for cardinal numbers from this theorem. (Contributed by NM, 4-Apr-2007.)
⊢ ((𝐴 ≈ 1o ∧ 𝐵 ≈ 1o) → ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐴 ∪ 𝐵) ≈ 2o))
Theorem	enpr2 10054	An unordered pair with distinct elements is equinumerous to ordinal two. This is a closed-form version of enpr2d 9054. (Contributed by FL, 17-Aug-2008.) Avoid ax-pow 5326, ax-un 7734. (Revised by BTernaryTau, 30-Dec-2024.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≈ 2o)
Theorem	pr2ne 10055	If an unordered pair has two elements, then they are different. (Contributed by FL, 14-Feb-2010.) Avoid ax-pow 5326, ax-un 7734. (Revised by BTernaryTau, 30-Dec-2024.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝐴, 𝐵} ≈ 2o ↔ 𝐴 ≠ 𝐵))
Theorem	prdom2 10056	An unordered pair has at most two elements. (Contributed by FL, 22-Feb-2011.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴, 𝐵} ≼ 2o)
Theorem	en2eqpr 10057	Building a set with two elements. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
⊢ ((𝐶 ≈ 2o ∧ 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐴 ≠ 𝐵 → 𝐶 = {𝐴, 𝐵}))
Theorem	en2eleq 10058	Express a set of pair cardinality as the unordered pair of a given element and the other element. (Contributed by Stefan O'Rear, 22-Aug-2015.)
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑃 = {𝑋, ∪ (𝑃 ∖ {𝑋})})
Theorem	en2other2 10059	Taking the other element twice in a pair gets back to the original element. (Contributed by Stefan O'Rear, 22-Aug-2015.)
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) = 𝑋)
Theorem	dif1card 10060	The cardinality of a nonempty finite set is one greater than the cardinality of the set with one element removed. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 2-Feb-2013.)
⊢ ((𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐴) → (card‘𝐴) = suc (card‘(𝐴 ∖ {𝑋})))
Theorem	leweon 10061*	Lexicographical order is a well-ordering of On × On. Proposition 7.56(1) of [TakeutiZaring] p. 54. Note that unlike r0weon 10062, this order is not set-like, as the preimage of ⟨1o, ∅⟩ is the proper class ({∅} × On). (Contributed by Mario Carneiro, 9-Mar-2013.)
⊢ 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st ‘𝑥) ∈ (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) ∈ (2nd ‘𝑦))))}    ⇒   ⊢ 𝐿 We (On × On)
Theorem	r0weon 10062*	A set-like well-ordering of the class of ordinal pairs. Proposition 7.58(1) of [TakeutiZaring] p. 54. (Contributed by Mario Carneiro, 7-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
⊢ 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st ‘𝑥) ∈ (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) ∈ (2nd ‘𝑦))))}    &   ⊢ 𝑅 = {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st ‘𝑧) ∪ (2nd ‘𝑧)) ∈ ((1st ‘𝑤) ∪ (2nd ‘𝑤)) ∨ (((1st ‘𝑧) ∪ (2nd ‘𝑧)) = ((1st ‘𝑤) ∪ (2nd ‘𝑤)) ∧ 𝑧𝐿𝑤)))}    ⇒   ⊢ (𝑅 We (On × On) ∧ 𝑅 Se (On × On))
Theorem	infxpenlem 10063*	Lemma for infxpen 10064. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
⊢ 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st ‘𝑥) ∈ (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) ∈ (2nd ‘𝑦))))}    &   ⊢ 𝑅 = {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st ‘𝑧) ∪ (2nd ‘𝑧)) ∈ ((1st ‘𝑤) ∪ (2nd ‘𝑤)) ∨ (((1st ‘𝑧) ∪ (2nd ‘𝑧)) = ((1st ‘𝑤) ∪ (2nd ‘𝑤)) ∧ 𝑧𝐿𝑤)))}    &   ⊢ 𝑄 = (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎)))    &   ⊢ (𝜑 ↔ ((𝑎 ∈ On ∧ ∀𝑚 ∈ 𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚 ∈ 𝑎 𝑚 ≺ 𝑎)))    &   ⊢ 𝑀 = ((1st ‘𝑤) ∪ (2nd ‘𝑤))    &   ⊢ 𝐽 = OrdIso(𝑄, (𝑎 × 𝑎))    ⇒   ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
Theorem	infxpen 10064	Every infinite ordinal is equinumerous to its Cartesian square. Proposition 10.39 of [TakeutiZaring] p. 94, whose proof we follow closely. The key idea is to show that the relation 𝑅 is a well-ordering of (On × On) with the additional property that 𝑅-initial segments of (𝑥 × 𝑥) (where 𝑥 is a limit ordinal) are of cardinality at most 𝑥. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
Theorem	xpomen 10065	The Cartesian product of omega (the set of ordinal natural numbers) with itself is equinumerous to omega. Exercise 1 of [Enderton] p. 133. (Contributed by NM, 23-Jul-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)
⊢ (ω × ω) ≈ ω
Theorem	xpct 10066	The cartesian product of two countable sets is countable. (Contributed by Thierry Arnoux, 24-Sep-2017.)
⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 × 𝐵) ≼ ω)
Theorem	infxpidm2 10067	Every infinite well-orderable set is equinumerous to its Cartesian square. This theorem provides the basis for infinite cardinal arithmetic. Proposition 10.40 of [TakeutiZaring] p. 95. See also infxpidm 10617. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
Theorem	infxpenc 10068*	A canonical version of infxpen 10064, by a completely different approach (although it uses infxpen 10064 via xpomen 10065). Using Cantor's normal form, we can show that 𝐴 ↑o 𝐵 respects equinumerosity (oef1o 9677), so that all the steps of (ω↑𝑊) · (ω↑𝑊) ≈ ω↑(2𝑊) ≈ (ω↑2)↑𝑊 ≈ ω↑𝑊 can be verified using bijections to do the ordinal commutations. (The assumption on 𝑁 can be satisfied using cnfcom3c 9685.) (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 7-Jul-2019.)
⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → ω ⊆ 𝐴)    &   ⊢ (𝜑 → 𝑊 ∈ (On ∖ 1o))    &   ⊢ (𝜑 → 𝐹:(ω ↑o 2o)–1-1-onto→ω)    &   ⊢ (𝜑 → (𝐹‘∅) = ∅)    &   ⊢ (𝜑 → 𝑁:𝐴–1-1-onto→(ω ↑o 𝑊))    &   ⊢ 𝐾 = (𝑦 ∈ {𝑥 ∈ ((ω ↑o 2o) ↑m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡( I ↾ 𝑊))))    &   ⊢ 𝐻 = (((ω CNF 𝑊) ∘ 𝐾) ∘ ◡((ω ↑o 2o) CNF 𝑊))    &   ⊢ 𝐿 = (𝑦 ∈ {𝑥 ∈ (ω ↑m (𝑊 ·o 2o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦 ∘ ◡(𝑌 ∘ ◡𝑋))))    &   ⊢ 𝑋 = (𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤))    &   ⊢ 𝑌 = (𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((2o ·o 𝑤) +o 𝑧))    &   ⊢ 𝐽 = (((ω CNF (2o ·o 𝑊)) ∘ 𝐿) ∘ ◡(ω CNF (𝑊 ·o 2o)))    &   ⊢ 𝑍 = (𝑥 ∈ (ω ↑o 𝑊), 𝑦 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑥) +o 𝑦))    &   ⊢ 𝑇 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝑁‘𝑥), (𝑁‘𝑦)⟩)    &   ⊢ 𝐺 = (◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇))    ⇒   ⊢ (𝜑 → 𝐺:(𝐴 × 𝐴)–1-1-onto→𝐴)
Theorem	infxpenc2lem1 10069*	Lemma for infxpenc2 10072. (Contributed by Mario Carneiro, 30-May-2015.)
⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → ∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))    &   ⊢ 𝑊 = (◡(𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘ran (𝑛‘𝑏))    ⇒   ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → (𝑊 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑊)))
Theorem	infxpenc2lem2 10070*	Lemma for infxpenc2 10072. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 7-Jul-2019.)
⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → ∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))    &   ⊢ 𝑊 = (◡(𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘ran (𝑛‘𝑏))    &   ⊢ (𝜑 → 𝐹:(ω ↑o 2o)–1-1-onto→ω)    &   ⊢ (𝜑 → (𝐹‘∅) = ∅)    &   ⊢ 𝐾 = (𝑦 ∈ {𝑥 ∈ ((ω ↑o 2o) ↑m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡( I ↾ 𝑊))))    &   ⊢ 𝐻 = (((ω CNF 𝑊) ∘ 𝐾) ∘ ◡((ω ↑o 2o) CNF 𝑊))    &   ⊢ 𝐿 = (𝑦 ∈ {𝑥 ∈ (ω ↑m (𝑊 ·o 2o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦 ∘ ◡(𝑌 ∘ ◡𝑋))))    &   ⊢ 𝑋 = (𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤))    &   ⊢ 𝑌 = (𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((2o ·o 𝑤) +o 𝑧))    &   ⊢ 𝐽 = (((ω CNF (2o ·o 𝑊)) ∘ 𝐿) ∘ ◡(ω CNF (𝑊 ·o 2o)))    &   ⊢ 𝑍 = (𝑥 ∈ (ω ↑o 𝑊), 𝑦 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑥) +o 𝑦))    &   ⊢ 𝑇 = (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ ⟨((𝑛‘𝑏)‘𝑥), ((𝑛‘𝑏)‘𝑦)⟩)    &   ⊢ 𝐺 = (◡(𝑛‘𝑏) ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇))    ⇒   ⊢ (𝜑 → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))
Theorem	infxpenc2lem3 10071*	Lemma for infxpenc2 10072. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 7-Jul-2019.)
⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → ∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))    &   ⊢ 𝑊 = (◡(𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘ran (𝑛‘𝑏))    &   ⊢ (𝜑 → 𝐹:(ω ↑o 2o)–1-1-onto→ω)    &   ⊢ (𝜑 → (𝐹‘∅) = ∅)    ⇒   ⊢ (𝜑 → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))
Theorem	infxpenc2 10072*	Existence form of infxpenc 10068. A "uniform" or "canonical" version of infxpen 10064, asserting the existence of a single function 𝑔 that simultaneously demonstrates product idempotence of all ordinals below a given bound. (Contributed by Mario Carneiro, 30-May-2015.)
⊢ (𝐴 ∈ On → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))
Theorem	iunmapdisj 10073*	The union ∪ 𝑛 ∈ 𝐶(𝐴 ↑m 𝑛) is a disjoint union. (Contributed by Mario Carneiro, 17-May-2015.) (Revised by NM, 16-Jun-2017.)
⊢ ∃*𝑛 ∈ 𝐶 𝐵 ∈ (𝐴 ↑m 𝑛)
Theorem	fseqenlem1 10074*	Lemma for fseqen 10077. (Contributed by Mario Carneiro, 17-May-2015.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐹:(𝐴 × 𝐴)–1-1-onto→𝐴)    &   ⊢ 𝐺 = seqω((𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴 ↑m suc 𝑛) ↦ ((𝑓‘(𝑥 ↾ 𝑛))𝐹(𝑥‘𝑛)))), {⟨∅, 𝐵⟩})    ⇒   ⊢ ((𝜑 ∧ 𝐶 ∈ ω) → (𝐺‘𝐶):(𝐴 ↑m 𝐶)–1-1→𝐴)
Theorem	fseqenlem2 10075*	Lemma for fseqen 10077. (Contributed by Mario Carneiro, 17-May-2015.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐹:(𝐴 × 𝐴)–1-1-onto→𝐴)    &   ⊢ 𝐺 = seqω((𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴 ↑m suc 𝑛) ↦ ((𝑓‘(𝑥 ↾ 𝑛))𝐹(𝑥‘𝑛)))), {⟨∅, 𝐵⟩})    &   ⊢ 𝐾 = (𝑦 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘) ↦ ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩)    ⇒   ⊢ (𝜑 → 𝐾:∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘)–1-1→(ω × 𝐴))
Theorem	fseqdom 10076*	One half of fseqen 10077. (Contributed by Mario Carneiro, 18-Nov-2014.)
⊢ (𝐴 ∈ 𝑉 → (ω × 𝐴) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
Theorem	fseqen 10077*	A set that is equinumerous to its Cartesian product is equinumerous to the set of finite sequences on it. (This can be proven more easily using some choice but this proof avoids it.) (Contributed by Mario Carneiro, 18-Nov-2014.)
⊢ (((𝐴 × 𝐴) ≈ 𝐴 ∧ 𝐴 ≠ ∅) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴))
Theorem	infpwfidom 10078	The collection of finite subsets of a set dominates the set. (We use the weaker sethood assumption (𝒫 𝐴 ∩ Fin) ∈ V because this theorem also implies that 𝐴 is a set if 𝒫 𝐴 ∩ Fin is.) (Contributed by Mario Carneiro, 17-May-2015.)
⊢ ((𝒫 𝐴 ∩ Fin) ∈ V → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
Theorem	dfac8alem 10079*	Lemma for dfac8a 10080. If the power set of a set has a choice function, then the set is numerable. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 5-Jan-2013.)
⊢ 𝐹 = recs(𝐺)    &   ⊢ 𝐺 = (𝑓 ∈ V ↦ (𝑔‘(𝐴 ∖ ran 𝑓)))    ⇒   ⊢ (𝐴 ∈ 𝐶 → (∃𝑔∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))
Theorem	dfac8a 10080*	Numeration theorem: every set with a choice function on its power set is numerable. With AC, this reduces to the statement that every set is numerable. Similar to Theorem 10.3 of [TakeutiZaring] p. 84. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 5-Jan-2013.)
⊢ (𝐴 ∈ 𝐵 → (∃ℎ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (ℎ‘𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))
Theorem	dfac8b 10081*	The well-ordering theorem: every numerable set is well-orderable. (Contributed by Mario Carneiro, 5-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ (𝐴 ∈ dom card → ∃𝑥 𝑥 We 𝐴)
Theorem	dfac8clem 10082*	Lemma for dfac8c 10083. (Contributed by Mario Carneiro, 10-Jan-2013.)
⊢ 𝐹 = (𝑠 ∈ (𝐴 ∖ {∅}) ↦ (℩𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ¬ 𝑏𝑟𝑎))    ⇒   ⊢ (𝐴 ∈ 𝐵 → (∃𝑟 𝑟 We ∪ 𝐴 → ∃𝑓∀𝑧 ∈ 𝐴 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
Theorem	dfac8c 10083*	If the union of a set is well-orderable, then the set has a choice function. (Contributed by Mario Carneiro, 5-Jan-2013.)
⊢ (𝐴 ∈ 𝐵 → (∃𝑟 𝑟 We ∪ 𝐴 → ∃𝑓∀𝑧 ∈ 𝐴 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
Theorem	ac10ct 10084*	A proof of the well-ordering theorem weth 10544, an Axiom of Choice equivalent, restricted to sets dominated by some ordinal (in particular finite sets and countable sets), proven in ZF without AC. (Contributed by Mario Carneiro, 5-Jan-2013.)
⊢ (∃𝑦 ∈ On 𝐴 ≼ 𝑦 → ∃𝑥 𝑥 We 𝐴)
Theorem	ween 10085*	A set is numerable iff it can be well-ordered. (Contributed by Mario Carneiro, 5-Jan-2013.)
⊢ (𝐴 ∈ dom card ↔ ∃𝑟 𝑟 We 𝐴)
Theorem	ac5num 10086*	A version of ac5b 10527 with the choice as a hypothesis. (Contributed by Mario Carneiro, 27-Aug-2015.)
⊢ ((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) → ∃𝑓(𝑓:𝐴⟶∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥))
Theorem	ondomen 10087	If a set is dominated by an ordinal, then it is numerable. (Contributed by Mario Carneiro, 5-Jan-2013.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ dom card)
Theorem	numdom 10088	A set dominated by a numerable set is numerable. (Contributed by Mario Carneiro, 28-Apr-2015.)
⊢ ((𝐴 ∈ dom card ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ dom card)
Theorem	ssnum 10089	A subset of a numerable set is numerable. (Contributed by Mario Carneiro, 28-Apr-2015.)
⊢ ((𝐴 ∈ dom card ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ dom card)
Theorem	onssnum 10090	All subsets of the ordinals are numerable. (Contributed by Mario Carneiro, 12-Feb-2013.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ⊆ On) → 𝐴 ∈ dom card)
Theorem	indcardi 10091*	Indirect strong induction on the cardinality of a finite or numerable set. (Contributed by Stefan O'Rear, 24-Aug-2015.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑇 ∈ dom card)    &   ⊢ ((𝜑 ∧ 𝑅 ≼ 𝑇 ∧ ∀𝑦(𝑆 ≺ 𝑅 → 𝜒)) → 𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))    &   ⊢ (𝑥 = 𝑦 → 𝑅 = 𝑆)    &   ⊢ (𝑥 = 𝐴 → 𝑅 = 𝑇)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	acnrcl 10092	Reverse closure for the choice set predicate. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ (𝑋 ∈ AC 𝐴 → 𝐴 ∈ V)
Theorem	acneq 10093	Equality theorem for the choice set function. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ (𝐴 = 𝐶 → AC 𝐴 = AC 𝐶)
Theorem	isacn 10094*	The property of being a choice set of length 𝐴. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑋 ∈ AC 𝐴 ↔ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)))
Theorem	acni 10095*	The property of being a choice set of length 𝐴. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ ((𝑋 ∈ AC 𝐴 ∧ 𝐹:𝐴⟶(𝒫 𝑋 ∖ {∅})) → ∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝐹‘𝑥))
Theorem	acni2 10096*	The property of being a choice set of length 𝐴. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ ((𝑋 ∈ AC 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐵 ⊆ 𝑋 ∧ 𝐵 ≠ ∅)) → ∃𝑔(𝑔:𝐴⟶𝑋 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵))
Theorem	acni3 10097*	The property of being a choice set of length 𝐴. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ (𝑦 = (𝑔‘𝑥) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝑋 ∈ AC 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑋 𝜑) → ∃𝑔(𝑔:𝐴⟶𝑋 ∧ ∀𝑥 ∈ 𝐴 𝜓))
Theorem	acnlem 10098*	Construct a mapping satisfying the consequent of isacn 10094. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥)) → ∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))
Theorem	numacn 10099	A well-orderable set has choice sequences of every length. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ (𝐴 ∈ 𝑉 → (𝑋 ∈ dom card → 𝑋 ∈ AC 𝐴))
Theorem	finacn 10100	Every set has finite choice sequences. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ (𝐴 ∈ Fin → AC 𝐴 = V)