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Theorem List for Metamath Proof Explorer - 9101-9200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremackbij2lem3 9101* Lemma for ackbij2 9103. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))    &   𝐺 = (𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥𝑦))))       (𝐴 ∈ ω → (rec(𝐺, ∅)‘𝐴) ⊆ (rec(𝐺, ∅)‘suc 𝐴))

Theoremackbij2lem4 9102* Lemma for ackbij2 9103. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))    &   𝐺 = (𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥𝑦))))       (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → (rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝐴))

Theoremackbij2 9103* 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‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))    &   𝐺 = (𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥𝑦))))    &   𝐻 = (rec(𝐺, ∅) “ ω)       𝐻: (𝑅1 “ ω)–1-1-onto→ω

Theoremr1om 9104 The set of hereditarily finite sets is countable. See ackbij2 9103 for an explicit bijection that works without Infinity. See also r1omALT 9636. (Contributed by Stefan O'Rear, 18-Nov-2014.)
(𝑅1‘ω) ≈ ω

Theoremfictb 9105 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.)
(𝐴𝐵 → (𝐴 ≼ ω ↔ (fi‘𝐴) ≼ ω))

2.6.11  Cofinality (without Axiom of Choice)

Theoremcflem 9106* 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.)
(𝐴𝑉 → ∃𝑥𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))

Theoremcfval 9107* 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) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})

Theoremcff 9108 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

Theoremcfub 9109* 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) ∧ (𝑦𝐴𝐴 𝑦))}

Theoremcflm 9110* Value of the cofinality function at a limit ordinal. Part of Definition of cofinality of [Enderton] p. 257. (Contributed by NM, 26-Apr-2004.)
((𝐴𝐵 ∧ Lim 𝐴) → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))})

Theoremcf0 9111 Value of the cofinality function at 0. Exercise 2 of [TakeutiZaring] p. 102. (Contributed by NM, 16-Apr-2004.)
(cf‘∅) = ∅

Theoremcardcf 9112 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)

Theoremcflecard 9113 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)

Theoremcfle 9114 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

Theoremcfon 9115 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

Theoremcfeq0 9116 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 = ∅))

Theoremcfsuc 9117 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)

Theoremcff1 9118* 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)–1-1𝐴 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤)))

Theoremcfflb 9119* If there is a cofinal map from 𝐵 to 𝐴, then 𝐵 is at least (cf‘𝐴). This theorem and cff1 9118 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:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (cf‘𝐴) ⊆ 𝐵))

Theoremcfval2 9120* Another expression for the cofinality function. (Contributed by Mario Carneiro, 28-Feb-2013.)
(𝐴 ∈ On → (cf‘𝐴) = 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤} (card‘𝑥))

Theoremcoflim 9121* A simpler expression for the cofinality predicate, at a limit ordinal. (Contributed by Mario Carneiro, 28-Feb-2013.)
((Lim 𝐴𝐵𝐴) → ( 𝐵 = 𝐴 ↔ ∀𝑥𝐴𝑦𝐵 𝑥𝑦))

Theoremcflim3 9122* 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))

Theoremcflim2 9123 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))

Theoremcfom 9124 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

Theoremcfss 9125* There is a cofinal subset of 𝐴 of cardinality (cf‘𝐴). (Contributed by Mario Carneiro, 24-Jun-2013.)
𝐴 ∈ V       (Lim 𝐴 → ∃𝑥(𝑥𝐴𝑥 ≈ (cf‘𝐴) ∧ 𝑥 = 𝐴))

Theoremcfslb 9126 Any cofinal subset of 𝐴 is at least as large as (cf‘𝐴). (Contributed by Mario Carneiro, 24-Jun-2013.)
𝐴 ∈ V       ((Lim 𝐴𝐵𝐴 𝐵 = 𝐴) → (cf‘𝐴) ≼ 𝐵)

Theoremcfslbn 9127 Any subset of 𝐴 smaller than its cofinality has union less than 𝐴. (This is the contrapositive to cfslb 9126.) (Contributed by Mario Carneiro, 24-Jun-2013.)
𝐴 ∈ V       ((Lim 𝐴𝐵𝐴𝐵 ≺ (cf‘𝐴)) → 𝐵𝐴)

Theoremcfslb2n 9128* 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 9126. 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 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → (𝐵 ≺ (cf‘𝐴) → 𝐵𝐴))

Theoremcofsmo 9129* 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 = {𝑦𝐵 ∣ ∀𝑤𝑦 (𝑓𝑤) ∈ (𝑓𝑦)}    &   𝐾 = {𝑥𝐵𝑧 ⊆ (𝑓𝑥)}    &   𝑂 = OrdIso( E , 𝐶)       ((Ord 𝐴𝐵 ∈ On) → (∃𝑓(𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → ∃𝑥 ∈ suc 𝐵𝑔(𝑔:𝑥𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣𝑥 𝑧 ⊆ (𝑔𝑣))))

Theoremcfsmolem 9130* Lemma for cfsmo 9131. (Contributed by Mario Carneiro, 28-Feb-2013.)
𝐹 = (𝑧 ∈ V ↦ ((𝑔‘dom 𝑧) ∪ 𝑡 ∈ dom 𝑧 suc (𝑧𝑡)))    &   𝐺 = (recs(𝐹) ↾ (cf‘𝐴))       (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤)))

Theoremcfsmo 9131* The map in cff1 9118 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 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤)))

Theoremcfcoflem 9132* Lemma for cfcof 9134, 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:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) → (cf‘𝐴) ⊆ (cf‘𝐵)))

Theoremcoftr 9133* 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 9134. (Contributed by Mario Carneiro, 16-Mar-2013.)
𝐻 = (𝑡𝐶 {𝑛𝐵 ∣ (𝑔𝑡) ⊆ (𝑓𝑛)})       (∃𝑓(𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) → (∃𝑔(𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤)) → ∃(:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤))))

Theoremcfcof 9134* 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:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (cf‘𝐴) = (cf‘𝐵)))

Theoremcfidm 9135 The cofinality function is idempotent. (Contributed by Mario Carneiro, 7-Mar-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)
(cf‘(cf‘𝐴)) = (cf‘𝐴)

Theoremalephsing 9136 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 8969). 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

Theoremsornom 9137* 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 ((𝐹𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) → 𝑅 Or ran 𝐹)

Syntaxcfin1a 9138 Extend class notation to include the class of Ia-finite sets.
class FinIa

Syntaxcfin2 9139 Extend class notation to include the class of II-finite sets.
class FinII

Syntaxcfin4 9140 Extend class notation to include the class of IV-finite sets.
class FinIV

Syntaxcfin3 9141 Extend class notation to include the class of III-finite sets.
class FinIII

Syntaxcfin5 9142 Extend class notation to include the class of V-finite sets.
class FinV

Syntaxcfin6 9143 Extend class notation to include the class of VI-finite sets.
class FinVI

Syntaxcfin7 9144 Extend class notation to include the class of VII-finite sets.
class FinVII

Definitiondf-fin1a 9145* 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 8001 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 ∨ (𝑥𝑦) ∈ Fin)}

Definitiondf-fin2 9146* 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) → 𝑦𝑦)}

Definitiondf-fin4 9147* 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(𝑦𝑥𝑦𝑥)}

Definitiondf-fin3 9148 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}

Definitiondf-fin5 9149 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))}

Definitiondf-fin6 9150 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 ≺ 2𝑜𝑥 ≺ (𝑥 × 𝑥))}

Definitiondf-fin7 9151* 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)𝑥𝑦}

Theoremisfin1a 9152* Definition of a Ia-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
(𝐴𝑉 → (𝐴 ∈ FinIa ↔ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ Fin ∨ (𝐴𝑦) ∈ Fin)))

Theoremfin1ai 9153 Property of a Ia-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
((𝐴 ∈ FinIa𝑋𝐴) → (𝑋 ∈ Fin ∨ (𝐴𝑋) ∈ Fin))

Theoremisfin2 9154* Definition of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
(𝐴𝑉 → (𝐴 ∈ FinII ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦)))

Theoremfin2i 9155 Property of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
(((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [] Or 𝐵)) → 𝐵𝐵)

Theoremisfin3 9156 Definition of a III-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
(𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV)

Theoremisfin4 9157* Definition of a IV-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
(𝐴𝑉 → (𝐴 ∈ FinIV ↔ ¬ ∃𝑦(𝑦𝐴𝑦𝐴)))

Theoremfin4i 9158 Infer that a set is IV-infinite. (Contributed by Stefan O'Rear, 16-May-2015.)
((𝑋𝐴𝑋𝐴) → ¬ 𝐴 ∈ FinIV)

Theoremisfin5 9159 Definition of a V-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
(𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 +𝑐 𝐴)))

Theoremisfin6 9160 Definition of a VI-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
(𝐴 ∈ FinVI ↔ (𝐴 ≺ 2𝑜𝐴 ≺ (𝐴 × 𝐴)))

Theoremisfin7 9161* Definition of a VII-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
(𝐴𝑉 → (𝐴 ∈ FinVII ↔ ¬ ∃𝑦 ∈ (On ∖ ω)𝐴𝑦))

Theoremsdom2en01 9162 A set with less than two elements has 0 or 1. (Contributed by Stefan O'Rear, 30-Oct-2014.)
(𝐴 ≺ 2𝑜 ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1𝑜))

Theoreminfpssrlem1 9163 Lemma for infpssr 9168. (Contributed by Stefan O'Rear, 30-Oct-2014.)
(𝜑𝐵𝐴)    &   (𝜑𝐹:𝐵1-1-onto𝐴)    &   (𝜑𝐶 ∈ (𝐴𝐵))    &   𝐺 = (rec(𝐹, 𝐶) ↾ ω)       (𝜑 → (𝐺‘∅) = 𝐶)

Theoreminfpssrlem2 9164 Lemma for infpssr 9168. (Contributed by Stefan O'Rear, 30-Oct-2014.)
(𝜑𝐵𝐴)    &   (𝜑𝐹:𝐵1-1-onto𝐴)    &   (𝜑𝐶 ∈ (𝐴𝐵))    &   𝐺 = (rec(𝐹, 𝐶) ↾ ω)       (𝑀 ∈ ω → (𝐺‘suc 𝑀) = (𝐹‘(𝐺𝑀)))

Theoreminfpssrlem3 9165 Lemma for infpssr 9168. (Contributed by Stefan O'Rear, 30-Oct-2014.)
(𝜑𝐵𝐴)    &   (𝜑𝐹:𝐵1-1-onto𝐴)    &   (𝜑𝐶 ∈ (𝐴𝐵))    &   𝐺 = (rec(𝐹, 𝐶) ↾ ω)       (𝜑𝐺:ω⟶𝐴)

Theoreminfpssrlem4 9166 Lemma for infpssr 9168. (Contributed by Stefan O'Rear, 30-Oct-2014.)
(𝜑𝐵𝐴)    &   (𝜑𝐹:𝐵1-1-onto𝐴)    &   (𝜑𝐶 ∈ (𝐴𝐵))    &   𝐺 = (rec(𝐹, 𝐶) ↾ ω)       ((𝜑𝑀 ∈ ω ∧ 𝑁𝑀) → (𝐺𝑀) ≠ (𝐺𝑁))

Theoreminfpssrlem5 9167 Lemma for infpssr 9168. (Contributed by Stefan O'Rear, 30-Oct-2014.)
(𝜑𝐵𝐴)    &   (𝜑𝐹:𝐵1-1-onto𝐴)    &   (𝜑𝐶 ∈ (𝐴𝐵))    &   𝐺 = (rec(𝐹, 𝐶) ↾ ω)       (𝜑 → (𝐴𝑉 → ω ≼ 𝐴))

Theoreminfpssr 9168 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.)
((𝑋𝐴𝑋𝐴) → ω ≼ 𝐴)

Theoremfin4en1 9169 Dedekind finite is a cardinal property. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)
(𝐴𝐵 → (𝐴 ∈ FinIV𝐵 ∈ FinIV))

Theoremssfin4 9170 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 ∈ FinIV𝐵𝐴) → 𝐵 ∈ FinIV)

Theoremdomfin4 9171 A set dominated by a Dedekind finite set is Dedekind finite. (Contributed by Mario Carneiro, 16-May-2015.)
((𝐴 ∈ FinIV𝐵𝐴) → 𝐵 ∈ FinIV)

Theoremominf4 9172 ω is Dedekind infinite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Proof shortened by Mario Carneiro, 16-May-2015.)
¬ ω ∈ FinIV

TheoreminfpssALT 9173* Alternate proof of infpss 9077, shorter but requiring Replacement (ax-rep 4804). (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(𝑥𝐴𝑥𝐴))

Theoremisfin4-2 9174 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.)
(𝐴𝑉 → (𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝐴))

Theoremisfin4-3 9175 Alternate definition of IV-finite sets: they are strictly dominated by their successors. (Thus, the proper subset referred to in isfin4 9157 can be assumed to be only a singleton smaller than the original.) (Contributed by Mario Carneiro, 18-May-2015.)
(𝐴 ∈ FinIV𝐴 ≺ (𝐴 +𝑐 1𝑜))

Theoremfin23lem7 9176* Lemma for isfin2-2 9179. 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.)
((𝐴𝑉𝐵 ⊆ 𝒫 𝐴𝐵 ≠ ∅) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝐴𝑥) ∈ 𝐵} ≠ ∅)

Theoremfin23lem11 9177* Lemma for isfin2-2 9179. (Contributed by Stefan O'Rear, 31-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)
(𝑧 = (𝐴𝑥) → (𝜓𝜒))    &   (𝑤 = (𝐴𝑣) → (𝜑𝜃))    &   ((𝑥𝐴𝑣𝐴) → (𝜒𝜃))       (𝐵 ⊆ 𝒫 𝐴 → (∃𝑥 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵}∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝜑 → ∃𝑧𝐵𝑣𝐵 ¬ 𝜓))

Theoremfin2i2 9178 A II-finite set contains minimal elements for every nonempty chain. (Contributed by Mario Carneiro, 16-May-2015.)
(((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [] Or 𝐵)) → 𝐵𝐵)

Theoremisfin2-2 9179* FinII expressed in terms of minimal elements. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Proof shortened by Mario Carneiro, 16-May-2015.)
(𝐴𝑉 → (𝐴 ∈ FinII ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦)))

Theoremssfin2 9180 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 ∈ FinII𝐵𝐴) → 𝐵 ∈ FinII)

Theoremenfin2i 9181 II-finiteness is a cardinal property. (Contributed by Mario Carneiro, 18-May-2015.)
(𝐴𝐵 → (𝐴 ∈ FinII𝐵 ∈ FinII))

Theoremfin23lem24 9182 Lemma for fin23 9249. 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 𝐴𝐵𝐴) ∧ (𝐶𝐵𝐷𝐵)) → ((𝐶𝐵) = (𝐷𝐵) ↔ 𝐶 = 𝐷))

Theoremfincssdom 9183 In a chain of finite sets, dominance and subset coincide. (Contributed by Stefan O'Rear, 8-Nov-2014.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵𝐵𝐴)) → (𝐴𝐵𝐴𝐵))

Theoremfin23lem25 9184 Lemma for fin23 9249. In a chain of finite sets, equinumerosity is equivalent to equality. (Contributed by Stefan O'Rear, 1-Nov-2014.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵𝐵𝐴)) → (𝐴𝐵𝐴 = 𝐵))

Theoremfin23lem26 9185* Lemma for fin23lem22 9187. (Contributed by Stefan O'Rear, 1-Nov-2014.)
(((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃𝑗𝑆 (𝑗𝑆) ≈ 𝑖)

Theoremfin23lem23 9186* Lemma for fin23lem22 9187. (Contributed by Stefan O'Rear, 1-Nov-2014.)
(((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃!𝑗𝑆 (𝑗𝑆) ≈ 𝑖)

Theoremfin23lem22 9187* Lemma for fin23 9249 but could be used elsewhere if we find a good name for it. Explicit construction of a bijection (actually an isomorphism, see fin23lem27 9188) between an infinite subset of ω and ω itself. (Contributed by Stefan O'Rear, 1-Nov-2014.)
𝐶 = (𝑖 ∈ ω ↦ (𝑗𝑆 (𝑗𝑆) ≈ 𝑖))       ((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) → 𝐶:ω–1-1-onto𝑆)

Theoremfin23lem27 9188* The mapping constructed in fin23lem22 9187 is in fact an isomorphism. (Contributed by Stefan O'Rear, 2-Nov-2014.)
𝐶 = (𝑖 ∈ ω ↦ (𝑗𝑆 (𝑗𝑆) ≈ 𝑖))       ((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) → 𝐶 Isom E , E (ω, 𝑆))

Theoremisfin3ds 9189* Property of a III-finite set (descending sequence version). (Contributed by Mario Carneiro, 16-May-2015.)
𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔𝑚 ω)(∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎𝑏) → ran 𝑎 ∈ ran 𝑎)}       (𝐴𝑉 → (𝐴𝐹 ↔ ∀𝑓 ∈ (𝒫 𝐴𝑚 ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓)))

Theoremssfin3ds 9190* A subset of a III-finite set is III-finite. (Contributed by Stefan O'Rear, 4-Nov-2014.)
𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔𝑚 ω)(∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎𝑏) → ran 𝑎 ∈ ran 𝑎)}       ((𝐴𝐹𝐵𝐴) → 𝐵𝐹)

Theoremfin23lem12 9191* The beginning of the proof that every II-finite set (every chain of subsets has a maximal element) is III-finite (has no denumerable collection of subsets).

This first section is dedicated to the construction of 𝑈 and its intersection. First, the value of 𝑈 at a successor. (Contributed by Stefan O'Rear, 1-Nov-2014.)

𝑈 = seq𝜔((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)       (𝐴 ∈ ω → (𝑈‘suc 𝐴) = if(((𝑡𝐴) ∩ (𝑈𝐴)) = ∅, (𝑈𝐴), ((𝑡𝐴) ∩ (𝑈𝐴))))

Theoremfin23lem13 9192* Lemma for fin23 9249. Each step of 𝑈 is a decrease. (Contributed by Stefan O'Rear, 1-Nov-2014.)
𝑈 = seq𝜔((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)       (𝐴 ∈ ω → (𝑈‘suc 𝐴) ⊆ (𝑈𝐴))

Theoremfin23lem14 9193* Lemma for fin23 9249. 𝑈 will never evolve to an empty set if it did not start with one. (Contributed by Stefan O'Rear, 1-Nov-2014.)
𝑈 = seq𝜔((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)       ((𝐴 ∈ ω ∧ ran 𝑡 ≠ ∅) → (𝑈𝐴) ≠ ∅)

Theoremfin23lem15 9194* Lemma for fin23 9249. 𝑈 is a monotone function. (Contributed by Stefan O'Rear, 1-Nov-2014.)
𝑈 = seq𝜔((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)       (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → (𝑈𝐴) ⊆ (𝑈𝐵))

Theoremfin23lem16 9195* Lemma for fin23 9249. 𝑈 ranges over the original set; in particular ran 𝑈 is a set, although we do not assume here that 𝑈 is. (Contributed by Stefan O'Rear, 1-Nov-2014.)
𝑈 = seq𝜔((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)        ran 𝑈 = ran 𝑡

Theoremfin23lem19 9196* Lemma for fin23 9249. The first set in 𝑈 to see an input set is either contained in it or disjoint from it. (Contributed by Stefan O'Rear, 1-Nov-2014.)
𝑈 = seq𝜔((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)       (𝐴 ∈ ω → ((𝑈‘suc 𝐴) ⊆ (𝑡𝐴) ∨ ((𝑈‘suc 𝐴) ∩ (𝑡𝐴)) = ∅))

Theoremfin23lem20 9197* Lemma for fin23 9249. 𝑋 is either contained in or disjoint from all input sets. (Contributed by Stefan O'Rear, 1-Nov-2014.)
𝑈 = seq𝜔((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)       (𝐴 ∈ ω → ( ran 𝑈 ⊆ (𝑡𝐴) ∨ ( ran 𝑈 ∩ (𝑡𝐴)) = ∅))

Theoremfin23lem17 9198* Lemma for fin23 9249. By ? Fin3DS ? , 𝑈 achieves its minimum (𝑋 in the synopsis above, but we will not be assigning a symbol here). TODO: Fix comment; math symbol Fin3DS does not exist. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
𝑈 = seq𝜔((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)    &   𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}       (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ran 𝑈 ∈ ran 𝑈)

Theoremfin23lem21 9199* Lemma for fin23 9249. 𝑋 is not empty. We only need here that 𝑡 has at least one set in its range besides ; the much stronger hypothesis here will serve as our induction hypothesis though. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 6-May-2015.)
𝑈 = seq𝜔((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)    &   𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}       (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ran 𝑈 ≠ ∅)

Theoremfin23lem28 9200* Lemma for fin23 9249. The residual is also one-to-one. This preserves the induction invariant. (Contributed by Stefan O'Rear, 2-Nov-2014.)
𝑈 = seq𝜔((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)    &   𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}    &   𝑃 = {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)}    &   𝑄 = (𝑤 ∈ ω ↦ (𝑥𝑃 (𝑥𝑃) ≈ 𝑤))    &   𝑅 = (𝑤 ∈ ω ↦ (𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤))    &   𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))       (𝑡:ω–1-1→V → 𝑍:ω–1-1→V)

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392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
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