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Theorem List for Metamath Proof Explorer - 9701-9800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-fin6 9701 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 9702* 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 9703* 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 9704 Property of a Ia-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
((𝐴 ∈ FinIa𝑋𝐴) → (𝑋 ∈ Fin ∨ (𝐴𝑋) ∈ Fin))
 
Theoremisfin2 9705* 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 9706 Property of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
(((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [] Or 𝐵)) → 𝐵𝐵)
 
Theoremisfin3 9707 Definition of a III-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
(𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV)
 
Theoremisfin4 9708* Definition of a IV-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
(𝐴𝑉 → (𝐴 ∈ FinIV ↔ ¬ ∃𝑦(𝑦𝐴𝑦𝐴)))
 
Theoremfin4i 9709 Infer that a set is IV-infinite. (Contributed by Stefan O'Rear, 16-May-2015.)
((𝑋𝐴𝑋𝐴) → ¬ 𝐴 ∈ FinIV)
 
Theoremisfin5 9710 Definition of a V-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
(𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴𝐴)))
 
Theoremisfin6 9711 Definition of a VI-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
(𝐴 ∈ FinVI ↔ (𝐴 ≺ 2o𝐴 ≺ (𝐴 × 𝐴)))
 
Theoremisfin7 9712* Definition of a VII-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
(𝐴𝑉 → (𝐴 ∈ FinVII ↔ ¬ ∃𝑦 ∈ (On ∖ ω)𝐴𝑦))
 
Theoremsdom2en01 9713 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 9714 Lemma for infpssr 9719. (Contributed by Stefan O'Rear, 30-Oct-2014.)
(𝜑𝐵𝐴)    &   (𝜑𝐹:𝐵1-1-onto𝐴)    &   (𝜑𝐶 ∈ (𝐴𝐵))    &   𝐺 = (rec(𝐹, 𝐶) ↾ ω)       (𝜑 → (𝐺‘∅) = 𝐶)
 
Theoreminfpssrlem2 9715 Lemma for infpssr 9719. (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 9716 Lemma for infpssr 9719. (Contributed by Stefan O'Rear, 30-Oct-2014.)
(𝜑𝐵𝐴)    &   (𝜑𝐹:𝐵1-1-onto𝐴)    &   (𝜑𝐶 ∈ (𝐴𝐵))    &   𝐺 = (rec(𝐹, 𝐶) ↾ ω)       (𝜑𝐺:ω⟶𝐴)
 
Theoreminfpssrlem4 9717 Lemma for infpssr 9719. (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 9718 Lemma for infpssr 9719. (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 9719 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 9720 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 9721 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 9722 A set dominated by a Dedekind finite set is Dedekind finite. (Contributed by Mario Carneiro, 16-May-2015.)
((𝐴 ∈ FinIV𝐵𝐴) → 𝐵 ∈ FinIV)
 
Theoremominf4 9723 ω is Dedekind infinite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Proof shortened by Mario Carneiro, 16-May-2015.)
¬ ω ∈ FinIV
 
TheoreminfpssALT 9724* Alternate proof of infpss 9628, shorter but requiring Replacement (ax-rep 5182). (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 9725 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))
 
Theoremisfin4p1 9726 Alternate definition of IV-finite sets: they are strictly dominated by their successors. (Thus, the proper subset referred to in isfin4 9708 can be assumed to be only a singleton smaller than the original.) (Contributed by Mario Carneiro, 18-May-2015.)
(𝐴 ∈ FinIV𝐴 ≺ (𝐴 ⊔ 1o))
 
Theoremfin23lem7 9727* Lemma for isfin2-2 9730. 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 9728* Lemma for isfin2-2 9730. (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 9729 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 9730* 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 9731 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 9732 II-finiteness is a cardinal property. (Contributed by Mario Carneiro, 18-May-2015.)
(𝐴𝐵 → (𝐴 ∈ FinII𝐵 ∈ FinII))
 
Theoremfin23lem24 9733 Lemma for fin23 9800. 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 9734 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 9735 Lemma for fin23 9800. 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 9736* Lemma for fin23lem22 9738. (Contributed by Stefan O'Rear, 1-Nov-2014.)
(((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃𝑗𝑆 (𝑗𝑆) ≈ 𝑖)
 
Theoremfin23lem23 9737* Lemma for fin23lem22 9738. (Contributed by Stefan O'Rear, 1-Nov-2014.)
(((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃!𝑗𝑆 (𝑗𝑆) ≈ 𝑖)
 
Theoremfin23lem22 9738* Lemma for fin23 9800 but could be used elsewhere if we find a good name for it. Explicit construction of a bijection (actually an isomorphism, see fin23lem27 9739) 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 9739* The mapping constructed in fin23lem22 9738 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 9740* 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 9741* 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 9742* 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 9743* Lemma for fin23 9800. 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 9744* Lemma for fin23 9800. 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 9745* Lemma for fin23 9800. 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 9746* Lemma for fin23 9800. 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 9747* Lemma for fin23 9800. 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 9748* Lemma for fin23 9800. 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 9749* Lemma for fin23 9800. 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 9750* Lemma for fin23 9800. 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 9751* Lemma for fin23 9800. 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)
 
Theoremfin23lem29 9752* Lemma for fin23 9800. The residual is built from the same elements as the previous sequence. (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))        ran 𝑍 ran 𝑡
 
Theoremfin23lem30 9753* Lemma for fin23 9800. The residual is disjoint from the common set. (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))       (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅)
 
Theoremfin23lem31 9754* Lemma for fin23 9800. The residual is has a strictly smaller range than the previous sequence. This will be iterated to build an unbounded chain. (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–1-1𝑉𝐺𝐹 ran 𝑡𝐺) → ran 𝑍 ran 𝑡)
 
Theoremfin23lem32 9755* Lemma for fin23 9800. Wrap the previous construction into a function to hide the hypotheses. (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))       (𝐺𝐹 → ∃𝑓𝑏((𝑏:ω–1-1→V ∧ ran 𝑏𝐺) → ((𝑓𝑏):ω–1-1→V ∧ ran (𝑓𝑏) ⊊ ran 𝑏)))
 
Theoremfin23lem33 9756* Lemma for fin23 9800. Discharge hypotheses. (Contributed by Stefan O'Rear, 2-Nov-2014.)
𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}       (𝐺𝐹 → ∃𝑓𝑏((𝑏:ω–1-1→V ∧ ran 𝑏𝐺) → ((𝑓𝑏):ω–1-1→V ∧ ran (𝑓𝑏) ⊊ ran 𝑏)))
 
Theoremfin23lem34 9757* Lemma for fin23 9800. Establish induction invariants on 𝑌 which parameterizes our contradictory chain of subsets. In this section, is the hypothetically assumed family of subsets, 𝑔 is the ground set, and 𝑖 is the induction function constructed in the previous section. (Contributed by Stefan O'Rear, 2-Nov-2014.)
𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}    &   (𝜑:ω–1-1→V)    &   (𝜑 ran 𝐺)    &   (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ran 𝑗𝐺) → ((𝑖𝑗):ω–1-1→V ∧ ran (𝑖𝑗) ⊊ ran 𝑗)))    &   𝑌 = (rec(𝑖, ) ↾ ω)       ((𝜑𝐴 ∈ ω) → ((𝑌𝐴):ω–1-1→V ∧ ran (𝑌𝐴) ⊆ 𝐺))
 
Theoremfin23lem35 9758* Lemma for fin23 9800. Strict order property of 𝑌. (Contributed by Stefan O'Rear, 2-Nov-2014.)
𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}    &   (𝜑:ω–1-1→V)    &   (𝜑 ran 𝐺)    &   (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ran 𝑗𝐺) → ((𝑖𝑗):ω–1-1→V ∧ ran (𝑖𝑗) ⊊ ran 𝑗)))    &   𝑌 = (rec(𝑖, ) ↾ ω)       ((𝜑𝐴 ∈ ω) → ran (𝑌‘suc 𝐴) ⊊ ran (𝑌𝐴))
 
Theoremfin23lem36 9759* Lemma for fin23 9800. Weak order property of 𝑌. (Contributed by Stefan O'Rear, 2-Nov-2014.)
𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}    &   (𝜑:ω–1-1→V)    &   (𝜑 ran 𝐺)    &   (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ran 𝑗𝐺) → ((𝑖𝑗):ω–1-1→V ∧ ran (𝑖𝑗) ⊊ ran 𝑗)))    &   𝑌 = (rec(𝑖, ) ↾ ω)       (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵𝐴𝜑)) → ran (𝑌𝐴) ⊆ ran (𝑌𝐵))
 
Theoremfin23lem38 9760* Lemma for fin23 9800. The contradictory chain has no minimum. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}    &   (𝜑:ω–1-1→V)    &   (𝜑 ran 𝐺)    &   (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ran 𝑗𝐺) → ((𝑖𝑗):ω–1-1→V ∧ ran (𝑖𝑗) ⊊ ran 𝑗)))    &   𝑌 = (rec(𝑖, ) ↾ ω)       (𝜑 → ¬ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)))
 
Theoremfin23lem39 9761* Lemma for fin23 9800. Thus, we have that 𝑔 could not have been in 𝐹 after all. (Contributed by Stefan O'Rear, 4-Nov-2014.)
𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}    &   (𝜑:ω–1-1→V)    &   (𝜑 ran 𝐺)    &   (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ran 𝑗𝐺) → ((𝑖𝑗):ω–1-1→V ∧ ran (𝑖𝑗) ⊊ ran 𝑗)))    &   𝑌 = (rec(𝑖, ) ↾ ω)       (𝜑 → ¬ 𝐺𝐹)
 
Theoremfin23lem40 9762* Lemma for fin23 9800. FinII sets satisfy the descending chain condition. (Contributed by Stefan O'Rear, 3-Nov-2014.)
𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}       (𝐴 ∈ FinII𝐴𝐹)
 
Theoremfin23lem41 9763* Lemma for fin23 9800. A set which satisfies the descending sequence condition must be III-finite. (Contributed by Stefan O'Rear, 2-Nov-2014.)
𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}       (𝐴𝐹𝐴 ∈ FinIII)
 
Theoremisf32lem1 9764* Lemma for isfin3-2 9778. Derive weak ordering property. (Contributed by Stefan O'Rear, 5-Nov-2014.)
(𝜑𝐹:ω⟶𝒫 𝐺)    &   (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥))    &   (𝜑 → ¬ ran 𝐹 ∈ ran 𝐹)       (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵𝐴𝜑)) → (𝐹𝐴) ⊆ (𝐹𝐵))
 
Theoremisf32lem2 9765* Lemma for isfin3-2 9778. Non-minimum implies that there is always another decrease. (Contributed by Stefan O'Rear, 5-Nov-2014.)
(𝜑𝐹:ω⟶𝒫 𝐺)    &   (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥))    &   (𝜑 → ¬ ran 𝐹 ∈ ran 𝐹)       ((𝜑𝐴 ∈ ω) → ∃𝑎 ∈ ω (𝐴𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹𝑎)))
 
Theoremisf32lem3 9766* Lemma for isfin3-2 9778. Being a chain, difference sets are disjoint (one case). (Contributed by Stefan O'Rear, 5-Nov-2014.)
(𝜑𝐹:ω⟶𝒫 𝐺)    &   (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥))    &   (𝜑 → ¬ ran 𝐹 ∈ ran 𝐹)       (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵𝐴𝜑)) → (((𝐹𝐴) ∖ (𝐹‘suc 𝐴)) ∩ ((𝐹𝐵) ∖ (𝐹‘suc 𝐵))) = ∅)
 
Theoremisf32lem4 9767* Lemma for isfin3-2 9778. Being a chain, difference sets are disjoint. (Contributed by Stefan O'Rear, 5-Nov-2014.)
(𝜑𝐹:ω⟶𝒫 𝐺)    &   (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥))    &   (𝜑 → ¬ ran 𝐹 ∈ ran 𝐹)       (((𝜑𝐴𝐵) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (((𝐹𝐴) ∖ (𝐹‘suc 𝐴)) ∩ ((𝐹𝐵) ∖ (𝐹‘suc 𝐵))) = ∅)
 
Theoremisf32lem5 9768* Lemma for isfin3-2 9778. There are infinite decrease points. (Contributed by Stefan O'Rear, 5-Nov-2014.)
(𝜑𝐹:ω⟶𝒫 𝐺)    &   (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥))    &   (𝜑 → ¬ ran 𝐹 ∈ ran 𝐹)    &   𝑆 = {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹𝑦)}       (𝜑 → ¬ 𝑆 ∈ Fin)
 
Theoremisf32lem6 9769* Lemma for isfin3-2 9778. Each K value is nonempty. (Contributed by Stefan O'Rear, 5-Nov-2014.)
(𝜑𝐹:ω⟶𝒫 𝐺)    &   (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥))    &   (𝜑 → ¬ ran 𝐹 ∈ ran 𝐹)    &   𝑆 = {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹𝑦)}    &   𝐽 = (𝑢 ∈ ω ↦ (𝑣𝑆 (𝑣𝑆) ≈ 𝑢))    &   𝐾 = ((𝑤𝑆 ↦ ((𝐹𝑤) ∖ (𝐹‘suc 𝑤))) ∘ 𝐽)       ((𝜑𝐴 ∈ ω) → (𝐾𝐴) ≠ ∅)
 
Theoremisf32lem7 9770* Lemma for isfin3-2 9778. Different K values are disjoint. (Contributed by Stefan O'Rear, 5-Nov-2014.)
(𝜑𝐹:ω⟶𝒫 𝐺)    &   (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥))    &   (𝜑 → ¬ ran 𝐹 ∈ ran 𝐹)    &   𝑆 = {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹𝑦)}    &   𝐽 = (𝑢 ∈ ω ↦ (𝑣𝑆 (𝑣𝑆) ≈ 𝑢))    &   𝐾 = ((𝑤𝑆 ↦ ((𝐹𝑤) ∖ (𝐹‘suc 𝑤))) ∘ 𝐽)       (((𝜑𝐴𝐵) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝐾𝐴) ∩ (𝐾𝐵)) = ∅)
 
Theoremisf32lem8 9771* Lemma for isfin3-2 9778. K sets are subsets of the base. (Contributed by Stefan O'Rear, 6-Nov-2014.)
(𝜑𝐹:ω⟶𝒫 𝐺)    &   (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥))    &   (𝜑 → ¬ ran 𝐹 ∈ ran 𝐹)    &   𝑆 = {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹𝑦)}    &   𝐽 = (𝑢 ∈ ω ↦ (𝑣𝑆 (𝑣𝑆) ≈ 𝑢))    &   𝐾 = ((𝑤𝑆 ↦ ((𝐹𝑤) ∖ (𝐹‘suc 𝑤))) ∘ 𝐽)       ((𝜑𝐴 ∈ ω) → (𝐾𝐴) ⊆ 𝐺)
 
Theoremisf32lem9 9772* Lemma for isfin3-2 9778. Construction of the onto function. (Contributed by Stefan O'Rear, 5-Nov-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
(𝜑𝐹:ω⟶𝒫 𝐺)    &   (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥))    &   (𝜑 → ¬ ran 𝐹 ∈ ran 𝐹)    &   𝑆 = {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹𝑦)}    &   𝐽 = (𝑢 ∈ ω ↦ (𝑣𝑆 (𝑣𝑆) ≈ 𝑢))    &   𝐾 = ((𝑤𝑆 ↦ ((𝐹𝑤) ∖ (𝐹‘suc 𝑤))) ∘ 𝐽)    &   𝐿 = (𝑡𝐺 ↦ (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠))))       (𝜑𝐿:𝐺onto→ω)
 
Theoremisf32lem10 9773* Lemma for isfin3-2 . Write in terms of weak dominance. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
(𝜑𝐹:ω⟶𝒫 𝐺)    &   (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥))    &   (𝜑 → ¬ ran 𝐹 ∈ ran 𝐹)    &   𝑆 = {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹𝑦)}    &   𝐽 = (𝑢 ∈ ω ↦ (𝑣𝑆 (𝑣𝑆) ≈ 𝑢))    &   𝐾 = ((𝑤𝑆 ↦ ((𝐹𝑤) ∖ (𝐹‘suc 𝑤))) ∘ 𝐽)    &   𝐿 = (𝑡𝐺 ↦ (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠))))       (𝜑 → (𝐺𝑉 → ω ≼* 𝐺))
 
Theoremisf32lem11 9774* Lemma for isfin3-2 9778. Remove hypotheses from isf32lem10 9773. (Contributed by Stefan O'Rear, 17-May-2015.)
((𝐺𝑉 ∧ (𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹𝑏) ∧ ¬ ran 𝐹 ∈ ran 𝐹)) → ω ≼* 𝐺)
 
Theoremisf32lem12 9775* Lemma for isfin3-2 9778. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}       (𝐺𝑉 → (¬ ω ≼* 𝐺𝐺𝐹))
 
Theoremisfin32i 9776 One half of isfin3-2 9778. (Contributed by Mario Carneiro, 3-Jun-2015.)
(𝐴 ∈ FinIII → ¬ ω ≼* 𝐴)
 
Theoremisf33lem 9777* Lemma for isfin3-3 9779. (Contributed by Stefan O'Rear, 17-May-2015.)
FinIII = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
 
Theoremisfin3-2 9778 Weakly Dedekind-infinite sets are exactly those which can be mapped onto ω. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
(𝐴𝑉 → (𝐴 ∈ FinIII ↔ ¬ ω ≼* 𝐴))
 
Theoremisfin3-3 9779* Weakly Dedekind-infinite sets are exactly those with an ω-indexed descending chain of subsets. (Contributed by Stefan O'Rear, 7-Nov-2014.)
(𝐴𝑉 → (𝐴 ∈ FinIII ↔ ∀𝑓 ∈ (𝒫 𝐴m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓)))
 
Theoremfin33i 9780* Inference from isfin3-3 9779. (This is actually a bit stronger than isfin3-3 9779 because it does not assume 𝐹 is a set and does not use the Axiom of Infinity either.) (Contributed by Mario Carneiro, 17-May-2015.)
((𝐴 ∈ FinIII𝐹:ω⟶𝒫 𝐴 ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥)) → ran 𝐹 ∈ ran 𝐹)
 
Theoremcompsscnvlem 9781* Lemma for compsscnv 9782. (Contributed by Mario Carneiro, 17-May-2015.)
((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦)))
 
Theoremcompsscnv 9782* Complementation on a power set lattice is an involution. (Contributed by Mario Carneiro, 17-May-2015.)
𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))       𝐹 = 𝐹
 
Theoremisf34lem1 9783* Lemma for isfin3-4 9793. (Contributed by Stefan O'Rear, 7-Nov-2014.)
𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))       ((𝐴𝑉𝑋𝐴) → (𝐹𝑋) = (𝐴𝑋))
 
Theoremisf34lem2 9784* Lemma for isfin3-4 9793. (Contributed by Stefan O'Rear, 7-Nov-2014.)
𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))       (𝐴𝑉𝐹:𝒫 𝐴⟶𝒫 𝐴)
 
Theoremcompssiso 9785* Complementation is an antiautomorphism on power set lattices. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))       (𝐴𝑉𝐹 Isom [] , [] (𝒫 𝐴, 𝒫 𝐴))
 
Theoremisf34lem3 9786* Lemma for isfin3-4 9793. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))       ((𝐴𝑉𝑋 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹𝑋)) = 𝑋)
 
Theoremcompss 9787* Express image under of the complementation isomorphism. (Contributed by Stefan O'Rear, 5-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))       (𝐹𝐺) = {𝑦 ∈ 𝒫 𝐴 ∣ (𝐴𝑦) ∈ 𝐺}
 
Theoremisf34lem4 9788* Lemma for isfin3-4 9793. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))       ((𝐴𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴𝑋 ≠ ∅)) → (𝐹 𝑋) = (𝐹𝑋))
 
Theoremisf34lem5 9789* Lemma for isfin3-4 9793. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))       ((𝐴𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴𝑋 ≠ ∅)) → (𝐹 𝑋) = (𝐹𝑋))
 
Theoremisf34lem7 9790* Lemma for isfin3-4 9793. (Contributed by Stefan O'Rear, 7-Nov-2014.)
𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))       ((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)) → ran 𝐺 ∈ ran 𝐺)
 
Theoremisf34lem6 9791* Lemma for isfin3-4 9793. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))       (𝐴𝑉 → (𝐴 ∈ FinIII ↔ ∀𝑓 ∈ (𝒫 𝐴m ω)(∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓)))
 
Theoremfin34i 9792* Inference from isfin3-4 9793. (Contributed by Mario Carneiro, 17-May-2015.)
((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑥 ∈ ω (𝐺𝑥) ⊆ (𝐺‘suc 𝑥)) → ran 𝐺 ∈ ran 𝐺)
 
Theoremisfin3-4 9793* Weakly Dedekind-infinite sets are exactly those with an ω-indexed ascending chain of subsets. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
(𝐴𝑉 → (𝐴 ∈ FinIII ↔ ∀𝑓 ∈ (𝒫 𝐴m ω)(∀𝑥 ∈ ω (𝑓𝑥) ⊆ (𝑓‘suc 𝑥) → ran 𝑓 ∈ ran 𝑓)))
 
Theoremfin11a 9794 Every I-finite set is Ia-finite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)
(𝐴 ∈ Fin → 𝐴 ∈ FinIa)
 
Theoremenfin1ai 9795 Ia-finiteness is a cardinal property. (Contributed by Mario Carneiro, 18-May-2015.)
(𝐴𝐵 → (𝐴 ∈ FinIa𝐵 ∈ FinIa))
 
Theoremisfin1-2 9796 A set is finite in the usual sense iff the power set of its power set is Dedekind finite. (Contributed by Stefan O'Rear, 3-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
(𝐴 ∈ Fin ↔ 𝒫 𝒫 𝐴 ∈ FinIV)
 
Theoremisfin1-3 9797 A set is I-finite iff every system of subsets contains a maximal subset. Definition I of [Levy58] p. 2. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
(𝐴𝑉 → (𝐴 ∈ Fin ↔ [] Fr 𝒫 𝐴))
 
Theoremisfin1-4 9798 A set is I-finite iff every system of subsets contains a minimal subset. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
(𝐴𝑉 → (𝐴 ∈ Fin ↔ [] Fr 𝒫 𝐴))
 
Theoremdffin1-5 9799 Compact quantifier-free version of the standard definition df-fin 8502. (Contributed by Stefan O'Rear, 6-Jan-2015.)
Fin = ( ≈ “ ω)
 
Theoremfin23 9800 Every II-finite set (every chain of subsets has a maximal element) is III-finite (has no denumerable collection of subsets). The proof here is the only one I could find, from http://matwbn.icm.edu.pl/ksiazki/fm/fm6/fm619.pdf p.94 (writeup by Tarski, credited to Kuratowski). Translated into English and modern notation, the proof proceeds as follows (variables renamed for uniqueness):

Suppose for a contradiction that 𝐴 is a set which is II-finite but not III-finite.

For any countable sequence of distinct subsets 𝑇 of 𝐴, we can form a decreasing sequence of nonempty subsets (𝑈𝑇) by taking finite intersections of initial segments of 𝑇 while skipping over any element of 𝑇 which would cause the intersection to be empty.

By II-finiteness (as fin2i2 9729) this sequence contains its intersection, call it 𝑌; since by induction every subset in the sequence 𝑈 is nonempty, the intersection must be nonempty.

Suppose that an element 𝑋 of 𝑇 has nonempty intersection with 𝑌. Thus, said element has a nonempty intersection with the corresponding element of 𝑈, therefore it was used in the construction of 𝑈 and all further elements of 𝑈 are subsets of 𝑋, thus 𝑋 contains the 𝑌. That is, all elements of 𝑋 either contain 𝑌 or are disjoint from it.

Since there are only two cases, there must exist an infinite subset of 𝑇 which uniformly either contain 𝑌 or are disjoint from it. In the former case we can create an infinite set by subtracting 𝑌 from each element. In either case, call the result 𝑍; this is an infinite set of subsets of 𝐴, each of which is disjoint from 𝑌 and contained in the union of 𝑇; the union of 𝑍 is strictly contained in the union of 𝑇, because only the latter is a superset of the nonempty set 𝑌.

The preceding four steps may be iterated a countable number of times starting from the assumed denumerable set of subsets to produce a denumerable sequence 𝐵 of the 𝑇 sets from each stage. Great caution is required to avoid ax-dc 9857 here; in particular an effective version of the pigeonhole principle (for aleph-null pigeons and 2 holes) is required. Since a denumerable set of subsets is assumed to exist, we can conclude ω ∈ V without the axiom.

This 𝐵 sequence is strictly decreasing, thus it has no minimum, contradicting the first assumption. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)

(𝐴 ∈ FinII𝐴 ∈ FinIII)
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