P. 103 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 10201-10300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ackbij1lem12 10201*	Lemma for ackbij1 10208. (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 10202*	Lemma for ackbij1 10208. (Contributed by Stefan O'Rear, 18-Nov-2014.)
⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))    ⇒   ⊢ (𝐹‘∅) = ∅
Theorem	ackbij1lem14 10203*	Lemma for ackbij1 10208. (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 10204*	Lemma for ackbij1 10208. (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 10205*	Lemma for ackbij1 10208. (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 10206*	Lemma for ackbij1 10208. (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 10207*	Lemma for ackbij1 10208. (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 10208*	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 10209*	The Ackermann bijection, part 1b: the bijection from ackbij1 10208 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 10210*	Lemma for ackbij2 10213. (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 10211*	Lemma for ackbij2 10213. (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 10212*	Lemma for ackbij2 10213. (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 10213*	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 10214	The set of hereditarily finite sets is countable. See ackbij2 10213 for an explicit bijection that works without Infinity. See also r1omALT 10749. (Contributed by Stefan O'Rear, 18-Nov-2014.)
⊢ (𝑅1‘ω) ≈ ω
Theorem	fictb 10215	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))
2.6.16  Cofinality (without Axiom of Choice)
Theorem	cflem 10216*	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.) Avoid ax-11 2194. (Revised by BTernaryTau, 25-Jul-2025.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
Theorem	cflemOLD 10217*	Obsolete version of cflem 10216 as of 25-Jul-2025. (Contributed by NM, 24-Apr-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
Theorem	cfval 10218*	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 10219	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 10220*	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 10221*	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 10222	Value of the cofinality function at 0. Exercise 2 of [TakeutiZaring] p. 102. (Contributed by NM, 16-Apr-2004.)
⊢ (cf‘∅) = ∅
Theorem	cardcf 10223	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 10224	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 10225	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 10226	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.) Avoid ax-pow 5327 and ax-un 7722. (Revised by Umit Teoman Dogan, 10-Jun-2026.)
⊢ (cf‘𝐴) ∈ On
Theorem	cfonOLD 10227	Obsolete version of cfon 10226 as of 10-Jun-2026. (Contributed by Mario Carneiro, 9-Mar-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (cf‘𝐴) ∈ On
Theorem	cfeq0 10228	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 10229	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 10230*	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 10231*	If there is a cofinal map from 𝐵 to 𝐴, then 𝐵 is at least (cf‘𝐴). This theorem and cff1 10230 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 10232*	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 10233*	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 10234*	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 10235	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 10236	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 10237*	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 10238	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 10239	Any subset of 𝐴 smaller than its cofinality has union less than 𝐴. (This is the contrapositive to cfslb 10238.) (Contributed by Mario Carneiro, 24-Jun-2013.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≺ (cf‘𝐴)) → ∪ 𝐵 ∈ 𝐴)
Theorem	cfslb2n 10240*	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 10238. 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 10241*	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 10242*	Lemma for cfsmo 10243. (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 10243*	The map in cff1 10230 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 10244*	Lemma for cfcof 10246, 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 10245*	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 10246. (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 10246*	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 10247	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 10248	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 10080). Proposition 11.13 of [TakeutiZaring] p. 103. (Contributed by Mario Carneiro, 9-Mar-2013.)
⊢ (Lim 𝐴 → (cf‘(ℵ‘𝐴)) = (cf‘𝐴))
2.6.17  Eight inequivalent definitions of finite set
Theorem	sornom 10249*	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 10250	Extend class notation to include the class of Ia-finite sets.
class FinIa
Syntax	cfin2 10251	Extend class notation to include the class of II-finite sets.
class FinII
Syntax	cfin4 10252	Extend class notation to include the class of IV-finite sets.
class FinIV
Syntax	cfin3 10253	Extend class notation to include the class of III-finite sets.
class FinIII
Syntax	cfin5 10254	Extend class notation to include the class of V-finite sets.
class FinV
Syntax	cfin6 10255	Extend class notation to include the class of VI-finite sets.
class FinVI
Syntax	cfin7 10256	Extend class notation to include the class of VII-finite sets.
class FinVII
Definition	df-fin1a 10257*	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 8935 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 10258*	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 10259*	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 10260	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 10261	A set is V-finite iff it behaves finitely under ⊔. Definition V of [Levy58] p. 3. (Contributed by Stefan O'Rear, 12-Nov-2014.)
⊢ FinV = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥))}
Definition	df-fin6 10262	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 10263*	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 10264*	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 10265	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 10266*	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 10267	Property of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [⊊] Or 𝐵)) → ∪ 𝐵 ∈ 𝐵)
Theorem	isfin3 10268	Definition of a III-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
⊢ (𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV)
Theorem	isfin4 10269*	Definition of a IV-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIV ↔ ¬ ∃𝑦(𝑦 ⊊ 𝐴 ∧ 𝑦 ≈ 𝐴)))
Theorem	fin4i 10270	Infer that a set is IV-infinite. (Contributed by Stefan O'Rear, 16-May-2015.)
⊢ ((𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴) → ¬ 𝐴 ∈ FinIV)
Theorem	isfin5 10271	Definition of a V-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
⊢ (𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴)))
Theorem	isfin6 10272	Definition of a VI-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
⊢ (𝐴 ∈ FinVI ↔ (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴)))
Theorem	isfin7 10273*	Definition of a VII-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinVII ↔ ¬ ∃𝑦 ∈ (On ∖ ω)𝐴 ≈ 𝑦))
Theorem	sdom2en01 10274	A set with less than two elements has 0 or 1. (Contributed by Stefan O'Rear, 30-Oct-2014.)
⊢ (𝐴 ≺ 2o ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1o))
Theorem	infpssrlem1 10275	Lemma for infpssr 10280. (Contributed by Stefan O'Rear, 30-Oct-2014.)
⊢ (𝜑 → 𝐵 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴 ∖ 𝐵))    &   ⊢ 𝐺 = (rec(◡𝐹, 𝐶) ↾ ω)    ⇒   ⊢ (𝜑 → (𝐺‘∅) = 𝐶)
Theorem	infpssrlem2 10276	Lemma for infpssr 10280. (Contributed by Stefan O'Rear, 30-Oct-2014.)
⊢ (𝜑 → 𝐵 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴 ∖ 𝐵))    &   ⊢ 𝐺 = (rec(◡𝐹, 𝐶) ↾ ω)    ⇒   ⊢ (𝑀 ∈ ω → (𝐺‘suc 𝑀) = (◡𝐹‘(𝐺‘𝑀)))
Theorem	infpssrlem3 10277	Lemma for infpssr 10280. (Contributed by Stefan O'Rear, 30-Oct-2014.)
⊢ (𝜑 → 𝐵 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴 ∖ 𝐵))    &   ⊢ 𝐺 = (rec(◡𝐹, 𝐶) ↾ ω)    ⇒   ⊢ (𝜑 → 𝐺:ω⟶𝐴)
Theorem	infpssrlem4 10278	Lemma for infpssr 10280. (Contributed by Stefan O'Rear, 30-Oct-2014.)
⊢ (𝜑 → 𝐵 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴 ∖ 𝐵))    &   ⊢ 𝐺 = (rec(◡𝐹, 𝐶) ↾ ω)    ⇒   ⊢ ((𝜑 ∧ 𝑀 ∈ ω ∧ 𝑁 ∈ 𝑀) → (𝐺‘𝑀) ≠ (𝐺‘𝑁))
Theorem	infpssrlem5 10279	Lemma for infpssr 10280. (Contributed by Stefan O'Rear, 30-Oct-2014.)
⊢ (𝜑 → 𝐵 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴 ∖ 𝐵))    &   ⊢ 𝐺 = (rec(◡𝐹, 𝐶) ↾ ω)    ⇒   ⊢ (𝜑 → (𝐴 ∈ 𝑉 → ω ≼ 𝐴))
Theorem	infpssr 10280	Dedekind infinity implies existence of a denumerable subset: take a single point witnessing the proper subset relation and iterate the embedding. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)
⊢ ((𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴) → ω ≼ 𝐴)
Theorem	fin4en1 10281	Dedekind finite is a cardinal property. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ FinIV → 𝐵 ∈ FinIV))
Theorem	ssfin4 10282	Dedekind finite sets have Dedekind finite subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 6-May-2015.) (Revised by Mario Carneiro, 16-May-2015.)
⊢ ((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ FinIV)
Theorem	domfin4 10283	A set dominated by a Dedekind finite set is Dedekind finite. (Contributed by Mario Carneiro, 16-May-2015.)
⊢ ((𝐴 ∈ FinIV ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ FinIV)
Theorem	ominf4 10284	ω is Dedekind infinite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Proof shortened by Mario Carneiro, 16-May-2015.)
⊢ ¬ ω ∈ FinIV
Theorem	infpssALT 10285*	Alternate proof of infpss 10187, shorter but requiring Replacement (ax-rep 5232). (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (ω ≼ 𝐴 → ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴))
Theorem	isfin4-2 10286	Alternate definition of IV-finite sets: they lack a denumerable subset. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝐴))
Theorem	isfin4p1 10287	Alternate definition of IV-finite sets: they are strictly dominated by their successors. (Thus, the proper subset referred to in isfin4 10269 can be assumed to be only a singleton smaller than the original.) (Contributed by Mario Carneiro, 18-May-2015.)
⊢ (𝐴 ∈ FinIV ↔ 𝐴 ≺ (𝐴 ⊔ 1o))
Theorem	fin23lem7 10288*	Lemma for isfin2-2 10291. The componentwise complement of a nonempty collection of sets is nonempty. (Contributed by Stefan O'Rear, 31-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴 ∧ 𝐵 ≠ ∅) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑥) ∈ 𝐵} ≠ ∅)
Theorem	fin23lem11 10289*	Lemma for isfin2-2 10291. (Contributed by Stefan O'Rear, 31-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)
⊢ (𝑧 = (𝐴 ∖ 𝑥) → (𝜓 ↔ 𝜒))    &   ⊢ (𝑤 = (𝐴 ∖ 𝑣) → (𝜑 ↔ 𝜃))    &   ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑣 ⊆ 𝐴) → (𝜒 ↔ 𝜃))    ⇒   ⊢ (𝐵 ⊆ 𝒫 𝐴 → (∃𝑥 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵}∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝜑 → ∃𝑧 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ¬ 𝜓))
Theorem	fin2i2 10290	A II-finite set contains minimal elements for every nonempty chain. (Contributed by Mario Carneiro, 16-May-2015.)
⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [⊊] Or 𝐵)) → ∩ 𝐵 ∈ 𝐵)
Theorem	isfin2-2 10291*	FinII expressed in terms of minimal elements. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Proof shortened by Mario Carneiro, 16-May-2015.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinII ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦)))
Theorem	ssfin2 10292	A subset of a II-finite set is II-finite. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 16-May-2015.)
⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ FinII)
Theorem	enfin2i 10293	II-finiteness is a cardinal property. (Contributed by Mario Carneiro, 18-May-2015.)
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ FinII → 𝐵 ∈ FinII))
Theorem	fin23lem24 10294	Lemma for fin23 10361. In a class of ordinals, each element is fully identified by those of its predecessors which also belong to the class. (Contributed by Stefan O'Rear, 1-Nov-2014.)
⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → ((𝐶 ∩ 𝐵) = (𝐷 ∩ 𝐵) ↔ 𝐶 = 𝐷))
Theorem	fincssdom 10295	In a chain of finite sets, dominance and subset coincide. (Contributed by Stefan O'Rear, 8-Nov-2014.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) → (𝐴 ≼ 𝐵 ↔ 𝐴 ⊆ 𝐵))
Theorem	fin23lem25 10296	Lemma for fin23 10361. In a chain of finite sets, equinumerosity is equivalent to equality. (Contributed by Stefan O'Rear, 1-Nov-2014.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) → (𝐴 ≈ 𝐵 ↔ 𝐴 = 𝐵))
Theorem	fin23lem26 10297*	Lemma for fin23lem22 10299. (Contributed by Stefan O'Rear, 1-Nov-2014.)
⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖)
Theorem	fin23lem23 10298*	Lemma for fin23lem22 10299. (Contributed by Stefan O'Rear, 1-Nov-2014.)
⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃!𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖)
Theorem	fin23lem22 10299*	Lemma for fin23 10361 but could be used elsewhere if we find a good name for it. Explicit construction of a bijection (actually an isomorphism, see fin23lem27 10300) between an infinite subset of ω and ω itself. (Contributed by Stefan O'Rear, 1-Nov-2014.)
⊢ 𝐶 = (𝑖 ∈ ω ↦ (℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖))    ⇒   ⊢ ((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) → 𝐶:ω–1-1-onto→𝑆)
Theorem	fin23lem27 10300*	The mapping constructed in fin23lem22 10299 is in fact an isomorphism. (Contributed by Stefan O'Rear, 2-Nov-2014.)
⊢ 𝐶 = (𝑖 ∈ ω ↦ (℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖))    ⇒   ⊢ ((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) → 𝐶 Isom E , E (ω, 𝑆))