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	cdalepw 8901	If 𝐴 is idempotent under cardinal sum and 𝐵 is dominated by the power set of 𝐴, then so is the cardinal sum of 𝐴 and 𝐵. (Contributed by Mario Carneiro, 15-May-2015.)
⊢ (((𝐴 +𝑐 𝐴) ≈ 𝐴 ∧ 𝐵 ≼ 𝒫 𝐴) → (𝐴 +𝑐 𝐵) ≼ 𝒫 𝐴)
Theorem	onacda 8902	The cardinal and ordinal sums are always equinumerous. (Contributed by Mario Carneiro, 6-Feb-2013.) (Revised by Mario Carneiro, 30-May-2015.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) ≈ (𝐴 +𝑐 𝐵))
Theorem	cardacda 8903	The cardinal sum is equinumerous to an ordinal sum of the cardinals. (Contributed by Mario Carneiro, 6-Feb-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 +𝑐 𝐵) ≈ ((card‘𝐴) +𝑜 (card‘𝐵)))
Theorem	cdanum 8904	The cardinal sum of two numerable sets is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 +𝑐 𝐵) ∈ dom card)
Theorem	unnum 8905	The union of two numerable sets is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ∪ 𝐵) ∈ dom card)
Theorem	nnacda 8906	The cardinal and ordinal sums of finite ordinals are equal. (Contributed by Paul Chapman, 11-Apr-2009.) (Revised by Mario Carneiro, 6-Feb-2013.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (card‘(𝐴 +𝑐 𝐵)) = (𝐴 +𝑜 𝐵))
Theorem	ficardun 8907	The cardinality of the union of disjoint, finite sets is the ordinal sum of their cardinalities. (Contributed by Paul Chapman, 5-Jun-2009.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘(𝐴 ∪ 𝐵)) = ((card‘𝐴) +𝑜 (card‘𝐵)))
Theorem	ficardun2 8908	The cardinality of the union of finite sets is at most the ordinal sum of their cardinalities. (Contributed by Mario Carneiro, 5-Feb-2013.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴 ∪ 𝐵)) ⊆ ((card‘𝐴) +𝑜 (card‘𝐵)))
Theorem	pwsdompw 8909*	Lemma for domtriom 9148. This is the equinumerosity version of the algebraic identity Σ𝑘 ∈ 𝑛(2↑𝑘) = (2↑𝑛) − 1. (Contributed by Mario Carneiro, 7-Feb-2013.)
⊢ ((𝑛 ∈ ω ∧ ∀𝑘 ∈ suc 𝑛(𝐵‘𝑘) ≈ 𝒫 𝑘) → ∪ 𝑘 ∈ 𝑛 (𝐵‘𝑘) ≺ (𝐵‘𝑛))
Theorem	unctb 8910	The union of two countable sets is countable. (Contributed by FL, 25-Aug-2006.) (Proof shortened by Mario Carneiro, 30-Apr-2015.)
⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 ∪ 𝐵) ≼ ω)
Theorem	infcdaabs 8911	Absorption law for addition to an infinite cardinal. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴) → (𝐴 +𝑐 𝐵) ≈ 𝐴)
Theorem	infunabs 8912	An infinite set is equinumerous to its union with a smaller one. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴) → (𝐴 ∪ 𝐵) ≈ 𝐴)
Theorem	infcda 8913	The sum of two cardinal numbers is their maximum, if one of them is infinite. Proposition 10.41 of [TakeutiZaring] p. 95. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 +𝑐 𝐵) ≈ (𝐴 ∪ 𝐵))
Theorem	infdif 8914	The cardinality of an infinite set does not change after subtracting a strictly smaller one. Example in [Enderton] p. 164. (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ∖ 𝐵) ≈ 𝐴)
Theorem	infdif2 8915	Cardinality ordering for an infinite class difference. (Contributed by NM, 24-Mar-2007.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ((𝐴 ∖ 𝐵) ≼ 𝐵 ↔ 𝐴 ≼ 𝐵))
Theorem	infxpdom 8916	Dominance law for multiplication with an infinite cardinal. (Contributed by NM, 26-Mar-2006.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴) → (𝐴 × 𝐵) ≼ 𝐴)
Theorem	infxpabs 8917	Absorption law for multiplication with an infinite cardinal. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵 ≼ 𝐴)) → (𝐴 × 𝐵) ≈ 𝐴)
Theorem	infunsdom1 8918	The union of two sets that are strictly dominated by the infinite set 𝑋 is also dominated by 𝑋. This version of infunsdom 8919 assumes additionally that 𝐴 is the smaller of the two. (Contributed by Mario Carneiro, 14-Dec-2013.) (Revised by Mario Carneiro, 3-May-2015.)
⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) → (𝐴 ∪ 𝐵) ≺ 𝑋)
Theorem	infunsdom 8919	The union of two sets that are strictly dominated by the infinite set 𝑋 is also strictly dominated by 𝑋. (Contributed by Mario Carneiro, 3-May-2015.)
⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≺ 𝑋 ∧ 𝐵 ≺ 𝑋)) → (𝐴 ∪ 𝐵) ≺ 𝑋)
Theorem	infxp 8920	Absorption law for multiplication with an infinite cardinal. Equivalent to Proposition 10.41 of [TakeutiZaring] p. 95. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐴 × 𝐵) ≈ (𝐴 ∪ 𝐵))
Theorem	pwcdadom 8921	A property of dominance over a powerset, and a main lemma for gchac 9382. Similar to Lemma 2.3 of [KanamoriPincus] p. 420. (Contributed by Mario Carneiro, 15-May-2015.)
⊢ (𝒫 (𝐴 +𝑐 𝐴) ≼ (𝐴 +𝑐 𝐵) → 𝒫 𝐴 ≼ 𝐵)
Theorem	infpss 8922*	Every infinite set has an equinumerous proper subset, proved without AC or Infinity. Exercise 7 of [TakeutiZaring] p. 91. See also infpssALT 9018. (Contributed by NM, 23-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ (ω ≼ 𝐴 → ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴))
Theorem	infmap2 8923*	An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. Although this version of infmap 9277 avoids the axiom of choice, it requires the powerset of an infinite set to be well-orderable and so is usually not applicable. (Contributed by NM, 1-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ ((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑𝑚 𝐵) ∈ dom card) → (𝐴 ↑𝑚 𝐵) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)})
2.6.10  The Ackermann bijection
Theorem	ackbij2lem1 8924	Lemma for ackbij2 8948. (Contributed by Stefan O'Rear, 18-Nov-2014.)
⊢ (𝐴 ∈ ω → 𝒫 𝐴 ⊆ (𝒫 ω ∩ Fin))
Theorem	ackbij1lem1 8925	Lemma for ackbij2 8948. (Contributed by Stefan O'Rear, 18-Nov-2014.)
⊢ (¬ 𝐴 ∈ 𝐵 → (𝐵 ∩ suc 𝐴) = (𝐵 ∩ 𝐴))
Theorem	ackbij1lem2 8926	Lemma for ackbij2 8948. (Contributed by Stefan O'Rear, 18-Nov-2014.)
⊢ (𝐴 ∈ 𝐵 → (𝐵 ∩ suc 𝐴) = ({𝐴} ∪ (𝐵 ∩ 𝐴)))
Theorem	ackbij1lem3 8927	Lemma for ackbij2 8948. (Contributed by Stefan O'Rear, 18-Nov-2014.)
⊢ (𝐴 ∈ ω → 𝐴 ∈ (𝒫 ω ∩ Fin))
Theorem	ackbij1lem4 8928	Lemma for ackbij2 8948. (Contributed by Stefan O'Rear, 19-Nov-2014.)
⊢ (𝐴 ∈ ω → {𝐴} ∈ (𝒫 ω ∩ Fin))
Theorem	ackbij1lem5 8929	Lemma for ackbij2 8948. (Contributed by Stefan O'Rear, 19-Nov-2014.)
⊢ (𝐴 ∈ ω → (card‘𝒫 suc 𝐴) = ((card‘𝒫 𝐴) +𝑜 (card‘𝒫 𝐴)))
Theorem	ackbij1lem6 8930	Lemma for ackbij2 8948. (Contributed by Stefan O'Rear, 18-Nov-2014.)
⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐴 ∪ 𝐵) ∈ (𝒫 ω ∩ Fin))
Theorem	ackbij1lem7 8931*	Lemma for ackbij1 8943. (Contributed by Stefan O'Rear, 21-Nov-2014.)
⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))    ⇒   ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘𝐴) = (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)))
Theorem	ackbij1lem8 8932*	Lemma for ackbij1 8943. (Contributed by Stefan O'Rear, 19-Nov-2014.)
⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))    ⇒   ⊢ (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴))
Theorem	ackbij1lem9 8933*	Lemma for ackbij1 8943. (Contributed by Stefan O'Rear, 19-Nov-2014.)
⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))    ⇒   ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹‘(𝐴 ∪ 𝐵)) = ((𝐹‘𝐴) +𝑜 (𝐹‘𝐵)))
Theorem	ackbij1lem10 8934*	Lemma for ackbij1 8943. (Contributed by Stefan O'Rear, 18-Nov-2014.)
⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))    ⇒   ⊢ 𝐹:(𝒫 ω ∩ Fin)⟶ω
Theorem	ackbij1lem11 8935*	Lemma for ackbij1 8943. (Contributed by Stefan O'Rear, 18-Nov-2014.)
⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))    ⇒   ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ (𝒫 ω ∩ Fin))
Theorem	ackbij1lem12 8936*	Lemma for ackbij1 8943. (Contributed by Stefan O'Rear, 18-Nov-2014.)
⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))    ⇒   ⊢ ((𝐵 ∈ (𝒫 ω ∩ Fin) ∧ 𝐴 ⊆ 𝐵) → (𝐹‘𝐴) ⊆ (𝐹‘𝐵))
Theorem	ackbij1lem13 8937*	Lemma for ackbij1 8943. (Contributed by Stefan O'Rear, 18-Nov-2014.)
⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))    ⇒   ⊢ (𝐹‘∅) = ∅
Theorem	ackbij1lem14 8938*	Lemma for ackbij1 8943. (Contributed by Stefan O'Rear, 18-Nov-2014.)
⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))    ⇒   ⊢ (𝐴 ∈ ω → (𝐹‘{𝐴}) = suc (𝐹‘𝐴))
Theorem	ackbij1lem15 8939*	Lemma for ackbij1 8943. (Contributed by Stefan O'Rear, 18-Nov-2014.)
⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))    ⇒   ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝑐 ∈ ω ∧ 𝑐 ∈ 𝐴 ∧ ¬ 𝑐 ∈ 𝐵)) → ¬ (𝐹‘(𝐴 ∩ suc 𝑐)) = (𝐹‘(𝐵 ∩ suc 𝑐)))
Theorem	ackbij1lem16 8940*	Lemma for ackbij1 8943. (Contributed by Stefan O'Rear, 18-Nov-2014.)
⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))    ⇒   ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ((𝐹‘𝐴) = (𝐹‘𝐵) → 𝐴 = 𝐵))
Theorem	ackbij1lem17 8941*	Lemma for ackbij1 8943. (Contributed by Stefan O'Rear, 18-Nov-2014.)
⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))    ⇒   ⊢ 𝐹:(𝒫 ω ∩ Fin)–1-1→ω
Theorem	ackbij1lem18 8942*	Lemma for ackbij1 8943. (Contributed by Stefan O'Rear, 18-Nov-2014.)
⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))    ⇒   ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc (𝐹‘𝐴))
Theorem	ackbij1 8943*	The Ackermann bijection, part 1: each natural number can be uniquely coded in binary as a finite set of natural numbers and conversely. (Contributed by Stefan O'Rear, 18-Nov-2014.)
⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))    ⇒   ⊢ 𝐹:(𝒫 ω ∩ Fin)–1-1-onto→ω
Theorem	ackbij1b 8944*	The Ackermann bijection, part 1b: the bijection from ackbij1 8943 restricts naturally to the powers of particular naturals. (Contributed by Stefan O'Rear, 18-Nov-2014.)
⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))    ⇒   ⊢ (𝐴 ∈ ω → (𝐹 “ 𝒫 𝐴) = (card‘𝒫 𝐴))
Theorem	ackbij2lem2 8945*	Lemma for ackbij2 8948. (Contributed by Stefan O'Rear, 18-Nov-2014.)
⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))    &   ⊢ 𝐺 = (𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥 “ 𝑦))))    ⇒   ⊢ (𝐴 ∈ ω → (rec(𝐺, ∅)‘𝐴):(𝑅1‘𝐴)–1-1-onto→(card‘(𝑅1‘𝐴)))
Theorem	ackbij2lem3 8946*	Lemma for ackbij2 8948. (Contributed by Stefan O'Rear, 18-Nov-2014.)
⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))    &   ⊢ 𝐺 = (𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥 “ 𝑦))))    ⇒   ⊢ (𝐴 ∈ ω → (rec(𝐺, ∅)‘𝐴) ⊆ (rec(𝐺, ∅)‘suc 𝐴))
Theorem	ackbij2lem4 8947*	Lemma for ackbij2 8948. (Contributed by Stefan O'Rear, 18-Nov-2014.)
⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))    &   ⊢ 𝐺 = (𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥 “ 𝑦))))    ⇒   ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝐴) → (rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝐴))
Theorem	ackbij2 8948*	The Ackermann bijection, part 2: hereditarily finite sets can be represented by recursive binary notation. (Contributed by Stefan O'Rear, 18-Nov-2014.)
⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))    &   ⊢ 𝐺 = (𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥 “ 𝑦))))    &   ⊢ 𝐻 = ∪ (rec(𝐺, ∅) “ ω)    ⇒   ⊢ 𝐻:∪ (𝑅1 “ ω)–1-1-onto→ω
Theorem	r1om 8949	The set of hereditarily finite sets is countable. See ackbij2 8948 for an explicit bijection that works without Infinity. See also r1omALT 9477. (Contributed by Stefan O'Rear, 18-Nov-2014.)
⊢ (𝑅1‘ω) ≈ ω
Theorem	fictb 8950	A set is countable iff its collection of finite intersections is countable. (Contributed by Jeff Hankins, 24-Aug-2009.) (Proof shortened by Mario Carneiro, 17-May-2015.)
⊢ (𝐴 ∈ 𝐵 → (𝐴 ≼ ω ↔ (fi‘𝐴) ≼ ω))
2.6.11  Cofinality (without Axiom of Choice)
Theorem	cflem 8951*	A lemma used to simplify cofinality computations, showing the existence of the cardinal of an unbounded subset of a set 𝐴. (Contributed by NM, 24-Apr-2004.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
Theorem	cfval 8952*	Value of the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). The cofinality of an ordinal number 𝐴 is the cardinality (size) of the smallest unbounded subset 𝑦 of the ordinal number. Unbounded means that for every member of 𝐴, there is a member of 𝑦 that is at least as large. Cofinality is a measure of how "reachable from below" an ordinal is. (Contributed by NM, 1-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
⊢ (𝐴 ∈ On → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
Theorem	cff 8953	Cofinality is a function on the class of ordinal numbers to the class of cardinal numbers. (Contributed by Mario Carneiro, 15-Sep-2013.)
⊢ cf:On⟶On
Theorem	cfub 8954*	An upper bound on cofinality. (Contributed by NM, 25-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
⊢ (cf‘𝐴) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))}
Theorem	cflm 8955*	Value of the cofinality function at a limit ordinal. Part of Definition of cofinality of [Enderton] p. 257. (Contributed by NM, 26-Apr-2004.)
⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))})
Theorem	cf0 8956	Value of the cofinality function at 0. Exercise 2 of [TakeutiZaring] p. 102. (Contributed by NM, 16-Apr-2004.)
⊢ (cf‘∅) = ∅
Theorem	cardcf 8957	Cofinality is a cardinal number. Proposition 11.11 of [TakeutiZaring] p. 103. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
⊢ (card‘(cf‘𝐴)) = (cf‘𝐴)
Theorem	cflecard 8958	Cofinality is bounded by the cardinality of its argument. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
⊢ (cf‘𝐴) ⊆ (card‘𝐴)
Theorem	cfle 8959	Cofinality is bounded by its argument. Exercise 1 of [TakeutiZaring] p. 102. (Contributed by NM, 26-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
⊢ (cf‘𝐴) ⊆ 𝐴
Theorem	cfon 8960	The cofinality of any set is an ordinal (although it only makes sense when 𝐴 is an ordinal). (Contributed by Mario Carneiro, 9-Mar-2013.)
⊢ (cf‘𝐴) ∈ On
Theorem	cfeq0 8961	Only the ordinal zero has cofinality zero. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 12-Feb-2013.)
⊢ (𝐴 ∈ On → ((cf‘𝐴) = ∅ ↔ 𝐴 = ∅))
Theorem	cfsuc 8962	Value of the cofinality function at a successor ordinal. Exercise 3 of [TakeutiZaring] p. 102. (Contributed by NM, 23-Apr-2004.) (Revised by Mario Carneiro, 12-Feb-2013.)
⊢ (𝐴 ∈ On → (cf‘suc 𝐴) = 1𝑜)
Theorem	cff1 8963*	There is always a map from (cf‘𝐴) to 𝐴 (this is a stronger condition than the definition, which only presupposes a map from some 𝑦 ≈ (cf‘𝐴). (Contributed by Mario Carneiro, 28-Feb-2013.)
⊢ (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)–1-1→𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤)))
Theorem	cfflb 8964*	If there is a cofinal map from 𝐵 to 𝐴, then 𝐵 is at least (cf‘𝐴). This theorem and cff1 8963 motivate the picture of (cf‘𝐴) as the greatest lower bound of the domain of cofinal maps into 𝐴. (Contributed by Mario Carneiro, 28-Feb-2013.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑓(𝑓:𝐵⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤)) → (cf‘𝐴) ⊆ 𝐵))
Theorem	cfval2 8965*	Another expression for the cofinality function. (Contributed by Mario Carneiro, 28-Feb-2013.)
⊢ (𝐴 ∈ On → (cf‘𝐴) = ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤} (card‘𝑥))
Theorem	coflim 8966*	A simpler expression for the cofinality predicate, at a limit ordinal. (Contributed by Mario Carneiro, 28-Feb-2013.)
⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∪ 𝐵 = 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))
Theorem	cflim3 8967*	Another expression for the cofinality function. (Contributed by Mario Carneiro, 28-Feb-2013.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (Lim 𝐴 → (cf‘𝐴) = ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥))
Theorem	cflim2 8968	The cofinality function is a limit ordinal iff its argument is. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (Lim 𝐴 ↔ Lim (cf‘𝐴))
Theorem	cfom 8969	Value of the cofinality function at omega (the set of natural numbers). Exercise 4 of [TakeutiZaring] p. 102. (Contributed by NM, 23-Apr-2004.) (Proof shortened by Mario Carneiro, 11-Jun-2015.)
⊢ (cf‘ω) = ω
Theorem	cfss 8970*	There is a cofinal subset of 𝐴 of cardinality (cf‘𝐴). (Contributed by Mario Carneiro, 24-Jun-2013.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (Lim 𝐴 → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ (cf‘𝐴) ∧ ∪ 𝑥 = 𝐴))
Theorem	cfslb 8971	Any cofinal subset of 𝐴 is at least as large as (cf‘𝐴). (Contributed by Mario Carneiro, 24-Jun-2013.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∪ 𝐵 = 𝐴) → (cf‘𝐴) ≼ 𝐵)
Theorem	cfslbn 8972	Any subset of 𝐴 smaller than its cofinality has union less than 𝐴. (This is the contrapositive to cfslb 8971.) (Contributed by Mario Carneiro, 24-Jun-2013.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≺ (cf‘𝐴)) → ∪ 𝐵 ∈ 𝐴)
Theorem	cfslb2n 8973*	Any small collection of small subsets of 𝐴 cannot have union 𝐴, where "small" means smaller than the cofinality. This is a stronger version of cfslb 8971. This is a common application of cofinality: under AC, (ℵ‘1) is regular, so it is not a countable union of countable sets. (Contributed by Mario Carneiro, 24-Jun-2013.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ((Lim 𝐴 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ⊆ 𝐴 ∧ 𝑥 ≺ (cf‘𝐴))) → (𝐵 ≺ (cf‘𝐴) → ∪ 𝐵 ≠ 𝐴))
Theorem	cofsmo 8974*	Any cofinal map implies the existence of a strictly monotone cofinal map with a domain no larger than the original. Proposition 11.7 of [TakeutiZaring] p. 101. (Contributed by Mario Carneiro, 20-Mar-2013.)
⊢ 𝐶 = {𝑦 ∈ 𝐵 ∣ ∀𝑤 ∈ 𝑦 (𝑓‘𝑤) ∈ (𝑓‘𝑦)}    &   ⊢ 𝐾 = ∩ {𝑥 ∈ 𝐵 ∣ 𝑧 ⊆ (𝑓‘𝑥)}    &   ⊢ 𝑂 = OrdIso( E , 𝐶)    ⇒   ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On) → (∃𝑓(𝑓:𝐵⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤)) → ∃𝑥 ∈ suc 𝐵∃𝑔(𝑔:𝑥⟶𝐴 ∧ Smo 𝑔 ∧ ∀𝑧 ∈ 𝐴 ∃𝑣 ∈ 𝑥 𝑧 ⊆ (𝑔‘𝑣))))
Theorem	cfsmolem 8975*	Lemma for cfsmo 8976. (Contributed by Mario Carneiro, 28-Feb-2013.)
⊢ 𝐹 = (𝑧 ∈ V ↦ ((𝑔‘dom 𝑧) ∪ ∪ 𝑡 ∈ dom 𝑧 suc (𝑧‘𝑡)))    &   ⊢ 𝐺 = (recs(𝐹) ↾ (cf‘𝐴))    ⇒   ⊢ (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤)))
Theorem	cfsmo 8976*	The map in cff1 8963 can be assumed to be a strictly monotone ordinal function without loss of generality. (Contributed by Mario Carneiro, 28-Feb-2013.)
⊢ (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤)))
Theorem	cfcoflem 8977*	Lemma for cfcof 8979, showing subset relation in one direction. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Dec-2014.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑓(𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) → (cf‘𝐴) ⊆ (cf‘𝐵)))
Theorem	coftr 8978*	If there is a cofinal map from 𝐵 to 𝐴 and another from 𝐶 to 𝐴, then there is also a cofinal map from 𝐶 to 𝐵. Proposition 11.9 of [TakeutiZaring] p. 102. A limited form of transitivity for the "cof" relation. This is really a lemma for cfcof 8979. (Contributed by Mario Carneiro, 16-Mar-2013.)
⊢ 𝐻 = (𝑡 ∈ 𝐶 ↦ ∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑡) ⊆ (𝑓‘𝑛)})    ⇒   ⊢ (∃𝑓(𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) → (∃𝑔(𝑔:𝐶⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐶 𝑧 ⊆ (𝑔‘𝑤)) → ∃ℎ(ℎ:𝐶⟶𝐵 ∧ ∀𝑠 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑠 ⊆ (ℎ‘𝑤))))
Theorem	cfcof 8979*	If there is a cofinal map from 𝐴 to 𝐵, then they have the same cofinality. This was used as Definition 11.1 of [TakeutiZaring] p. 100, who defines an equivalence relation cof (𝐴, 𝐵) and defines our cf(𝐵) as the minimum 𝐵 such that cof (𝐴, 𝐵). (Contributed by Mario Carneiro, 20-Mar-2013.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑓(𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤)) → (cf‘𝐴) = (cf‘𝐵)))
Theorem	cfidm 8980	The cofinality function is idempotent. (Contributed by Mario Carneiro, 7-Mar-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)
⊢ (cf‘(cf‘𝐴)) = (cf‘𝐴)
Theorem	alephsing 8981	The cofinality of a limit aleph is the same as the cofinality of its argument, so if (ℵ‘𝐴) < 𝐴, then (ℵ‘𝐴) is singular. Conversely, if (ℵ‘𝐴) is regular (i.e. weakly inaccessible), then (ℵ‘𝐴) = 𝐴, so 𝐴 has to be rather large (see alephfp 8814). Proposition 11.13 of [TakeutiZaring] p. 103. (Contributed by Mario Carneiro, 9-Mar-2013.)
⊢ (Lim 𝐴 → (cf‘(ℵ‘𝐴)) = (cf‘𝐴))
2.6.12  Eight inequivalent definitions of finite set
Theorem	sornom 8982*	The range of a single-step monotone function from ω into a partially ordered set is a chain. (Contributed by Stefan O'Rear, 3-Nov-2014.)
⊢ ((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹‘𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹‘𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) → 𝑅 Or ran 𝐹)
Syntax	cfin1a 8983	Extend class notation to include the class of Ia-finite sets.
class FinIa
Syntax	cfin2 8984	Extend class notation to include the class of II-finite sets.
class FinII
Syntax	cfin4 8985	Extend class notation to include the class of IV-finite sets.
class FinIV
Syntax	cfin3 8986	Extend class notation to include the class of III-finite sets.
class FinIII
Syntax	cfin5 8987	Extend class notation to include the class of V-finite sets.
class FinV
Syntax	cfin6 8988	Extend class notation to include the class of VI-finite sets.
class FinVI
Syntax	cfin7 8989	Extend class notation to include the class of VII-finite sets.
class FinVII
Definition	df-fin1a 8990*	A set is Ia-finite iff it is not the union of two I-infinite sets. Equivalent to definition Ia of [Levy58] p. 2. A I-infinite Ia-finite set is also known as an amorphous set. This is the second of Levy's eight definitions of finite set. Levy's I-finite is equivalent to our df-fin 7845 and not repeated here. These eight definitions are equivalent with Choice but strictly decreasing in strength in models where Choice fails; conversely, they provide a series of increasingly stronger notions of infiniteness. (Contributed by Stefan O'Rear, 12-Nov-2014.)
⊢ FinIa = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥 ∖ 𝑦) ∈ Fin)}
Definition	df-fin2 8991*	A set is II-finite (Tarski finite) iff every nonempty chain of subsets contains a maximum element. Definition II of [Levy58] p. 2. (Contributed by Stefan O'Rear, 12-Nov-2014.)
⊢ FinII = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝒫 𝑥((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦)}
Definition	df-fin4 8992*	A set is IV-finite (Dedekind finite) iff it has no equinumerous proper subset. Definition IV of [Levy58] p. 3. (Contributed by Stefan O'Rear, 12-Nov-2014.)
⊢ FinIV = {𝑥 ∣ ¬ ∃𝑦(𝑦 ⊊ 𝑥 ∧ 𝑦 ≈ 𝑥)}
Definition	df-fin3 8993	A set is III-finite (weakly Dedekind finite) iff its power set is Dedekind finite. Definition III of [Levy58] p. 2. (Contributed by Stefan O'Rear, 12-Nov-2014.)
⊢ FinIII = {𝑥 ∣ 𝒫 𝑥 ∈ FinIV}
Definition	df-fin5 8994	A set is V-finite iff it behaves finitely under +𝑐. Definition V of [Levy58] p. 3. (Contributed by Stefan O'Rear, 12-Nov-2014.)
⊢ FinV = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 +𝑐 𝑥))}
Definition	df-fin6 8995	A set is VI-finite iff it behaves finitely under ×. Definition VI of [Levy58] p. 4. (Contributed by Stefan O'Rear, 12-Nov-2014.)
⊢ FinVI = {𝑥 ∣ (𝑥 ≺ 2𝑜 ∨ 𝑥 ≺ (𝑥 × 𝑥))}
Definition	df-fin7 8996*	A set is VII-finite iff it cannot be infinitely well-ordered. Equivalent to definition VII of [Levy58] p. 4. (Contributed by Stefan O'Rear, 12-Nov-2014.)
⊢ FinVII = {𝑥 ∣ ¬ ∃𝑦 ∈ (On ∖ ω)𝑥 ≈ 𝑦}
Theorem	isfin1a 8997*	Definition of a Ia-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIa ↔ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ Fin ∨ (𝐴 ∖ 𝑦) ∈ Fin)))
Theorem	fin1ai 8998	Property of a Ia-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
⊢ ((𝐴 ∈ FinIa ∧ 𝑋 ⊆ 𝐴) → (𝑋 ∈ Fin ∨ (𝐴 ∖ 𝑋) ∈ Fin))
Theorem	isfin2 8999*	Definition of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinII ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦)))
Theorem	fin2i 9000	Property of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [⊊] Or 𝐵)) → ∪ 𝐵 ∈ 𝐵)