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Theorem List for Metamath Proof Explorer - 8901-9000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremssfin2 8901 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 8902 II-finiteness is a cardinal property. (Contributed by Mario Carneiro, 18-May-2015.)
(𝐴𝐵 → (𝐴 ∈ FinII𝐵 ∈ FinII))

Theoremfin23lem24 8903 Lemma for fin23 8970. 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 8904 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 8905 Lemma for fin23 8970. 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 8906* Lemma for fin23lem22 8908. (Contributed by Stefan O'Rear, 1-Nov-2014.)
(((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃𝑗𝑆 (𝑗𝑆) ≈ 𝑖)

Theoremfin23lem23 8907* Lemma for fin23lem22 8908. (Contributed by Stefan O'Rear, 1-Nov-2014.)
(((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃!𝑗𝑆 (𝑗𝑆) ≈ 𝑖)

Theoremfin23lem22 8908* Lemma for fin23 8970 but could be used elsewhere if we find a good name for it. Explicit construction of a bijection (actually an isomorphism, see fin23lem27 8909) 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 8909* The mapping constructed in fin23lem22 8908 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 8910* 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 8911* 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 8912* 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 8913* Lemma for fin23 8970. 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 8914* Lemma for fin23 8970. 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 8915* Lemma for fin23 8970. 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 8916* Lemma for fin23 8970. 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 8917* Lemma for fin23 8970. 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 8918* Lemma for fin23 8970. 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 8919* Lemma for fin23 8970. 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 8920* Lemma for fin23 8970. 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 8921* Lemma for fin23 8970. 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 8922* Lemma for fin23 8970. 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 8923* Lemma for fin23 8970. 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 8924* Lemma for fin23 8970. 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 8925* Lemma for fin23 8970. 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 8926* Lemma for fin23 8970. 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 8927* Lemma for fin23 8970. 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 8928* Lemma for fin23 8970. 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 8929* Lemma for fin23 8970. 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 8930* Lemma for fin23 8970. 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 8931* Lemma for fin23 8970. 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 8932* Lemma for fin23 8970. 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 8933* Lemma for fin23 8970. 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 8934* Lemma for isfin3-2 8948. 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 8935* Lemma for isfin3-2 8948. 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 8936* Lemma for isfin3-2 8948. 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 8937* Lemma for isfin3-2 8948. 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 8938* Lemma for isfin3-2 8948. 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 8939* Lemma for isfin3-2 8948. 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 8940* Lemma for isfin3-2 8948. 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 8941* Lemma for isfin3-2 8948. 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 8942* Lemma for isfin3-2 8948. 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 8943* 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 8944* Lemma for isfin3-2 8948. Remove hypotheses from isf32lem10 8943. (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 8945* Lemma for isfin3-2 8948. (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 8946 One half of isfin3-2 8948. (Contributed by Mario Carneiro, 3-Jun-2015.)
(𝐴 ∈ FinIII → ¬ ω ≼* 𝐴)

Theoremisf33lem 8947* Lemma for isfin3-3 8949. (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 8948 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 8949* 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 8950* Inference from isfin3-3 8949. (This is actually a bit stronger than isfin3-3 8949 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 8951* Lemma for compsscnv 8952. (Contributed by Mario Carneiro, 17-May-2015.)
((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦)))

Theoremcompsscnv 8952* Complementation on a power set lattice is an involution. (Contributed by Mario Carneiro, 17-May-2015.)
𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))       𝐹 = 𝐹

Theoremisf34lem1 8953* Lemma for isfin3-4 8963. (Contributed by Stefan O'Rear, 7-Nov-2014.)
𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))       ((𝐴𝑉𝑋𝐴) → (𝐹𝑋) = (𝐴𝑋))

Theoremisf34lem2 8954* Lemma for isfin3-4 8963. (Contributed by Stefan O'Rear, 7-Nov-2014.)
𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))       (𝐴𝑉𝐹:𝒫 𝐴⟶𝒫 𝐴)

Theoremcompssiso 8955* 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 8956* Lemma for isfin3-4 8963. (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 8957* 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 8958* Lemma for isfin3-4 8963. (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 8959* Lemma for isfin3-4 8963. (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 8960* Lemma for isfin3-4 8963. (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 8961* Lemma for isfin3-4 8963. (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 8962* Inference from isfin3-4 8963. (Contributed by Mario Carneiro, 17-May-2015.)
((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑥 ∈ ω (𝐺𝑥) ⊆ (𝐺‘suc 𝑥)) → ran 𝐺 ∈ ran 𝐺)

Theoremisfin3-4 8963* 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 8964 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 8965 Ia-finiteness is a cardinal property. (Contributed by Mario Carneiro, 18-May-2015.)
(𝐴𝐵 → (𝐴 ∈ FinIa𝐵 ∈ FinIa))

Theoremisfin1-2 8966 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 8967 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 8968 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 8969 Compact quantifier-free version of the standard definition df-fin 7721. (Contributed by Stefan O'Rear, 6-Jan-2015.)
Fin = ( ≈ “ ω)

Theoremfin23 8970 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 8899) 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 9027 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)

Theoremfin34 8971 Every III-finite set is IV-finite. (Contributed by Stefan O'Rear, 30-Oct-2014.)
(𝐴 ∈ FinIII𝐴 ∈ FinIV)

Theoremisfin5-2 8972 Alternate definition of V-finite which emphasizes the idempotent behavior of V-infinite sets. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)
(𝐴𝑉 → (𝐴 ∈ FinV ↔ ¬ (𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 +𝑐 𝐴))))

Theoremfin45 8973 Every IV-finite set is V-finite: if we can pack two copies of the set into itself, we can certainly leave space. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Proof shortened by Mario Carneiro, 18-May-2015.)
(𝐴 ∈ FinIV𝐴 ∈ FinV)

Theoremfin56 8974 Every V-finite set is VI-finite because multiplication dominates addition for cardinals. (Contributed by Stefan O'Rear, 29-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)
(𝐴 ∈ FinV𝐴 ∈ FinVI)

Theoremfin17 8975 Every I-finite set is VII-finite. (Contributed by Mario Carneiro, 17-May-2015.)
(𝐴 ∈ Fin → 𝐴 ∈ FinVII)

Theoremfin67 8976 Every VI-finite set is VII-finite. (Contributed by Stefan O'Rear, 29-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)
(𝐴 ∈ FinVI𝐴 ∈ FinVII)

Theoremisfin7-2 8977 A set is VII-finite iff it is non-well-orderable or finite. (Contributed by Mario Carneiro, 17-May-2015.)
(𝐴𝑉 → (𝐴 ∈ FinVII ↔ (𝐴 ∈ dom card → 𝐴 ∈ Fin)))

Theoremfin71num 8978 A well-orderable set is VII-finite iff it is I-finite. Thus, even without choice, on the class of well-orderable sets all eight definitions of finite set coincide. (Contributed by Mario Carneiro, 18-May-2015.)
(𝐴 ∈ dom card → (𝐴 ∈ FinVII𝐴 ∈ Fin))

Theoremdffin7-2 8979 Class form of isfin7-2 8977. (Contributed by Mario Carneiro, 17-May-2015.)
FinVII = (Fin ∪ (V ∖ dom card))

Theoremdfacfin7 8980 Axiom of Choice equivalent: the VII-finite sets are the same as I-finite sets. (Contributed by Mario Carneiro, 18-May-2015.)
(CHOICE ↔ FinVII = Fin)

Theoremfin1a2lem1 8981 Lemma for fin1a2 8996. (Contributed by Stefan O'Rear, 7-Nov-2014.)
𝑆 = (𝑥 ∈ On ↦ suc 𝑥)       (𝐴 ∈ On → (𝑆𝐴) = suc 𝐴)

Theoremfin1a2lem2 8982 Lemma for fin1a2 8996. (Contributed by Stefan O'Rear, 7-Nov-2014.)
𝑆 = (𝑥 ∈ On ↦ suc 𝑥)       𝑆:On–1-1→On

Theoremfin1a2lem3 8983 Lemma for fin1a2 8996. (Contributed by Stefan O'Rear, 7-Nov-2014.)
𝐸 = (𝑥 ∈ ω ↦ (2𝑜 ·𝑜 𝑥))       (𝐴 ∈ ω → (𝐸𝐴) = (2𝑜 ·𝑜 𝐴))

Theoremfin1a2lem4 8984 Lemma for fin1a2 8996. (Contributed by Stefan O'Rear, 7-Nov-2014.)
𝐸 = (𝑥 ∈ ω ↦ (2𝑜 ·𝑜 𝑥))       𝐸:ω–1-1→ω

Theoremfin1a2lem5 8985 Lemma for fin1a2 8996. (Contributed by Stefan O'Rear, 7-Nov-2014.)
𝐸 = (𝑥 ∈ ω ↦ (2𝑜 ·𝑜 𝑥))       (𝐴 ∈ ω → (𝐴 ∈ ran 𝐸 ↔ ¬ suc 𝐴 ∈ ran 𝐸))

Theoremfin1a2lem6 8986 Lemma for fin1a2 8996. Establish that ω can be broken into two equipollent pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.)
𝐸 = (𝑥 ∈ ω ↦ (2𝑜 ·𝑜 𝑥))    &   𝑆 = (𝑥 ∈ On ↦ suc 𝑥)       (𝑆 ↾ ran 𝐸):ran 𝐸1-1-onto→(ω ∖ ran 𝐸)

Theoremfin1a2lem7 8987* Lemma for fin1a2 8996. Split a III-infinite set in two pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.)
𝐸 = (𝑥 ∈ ω ↦ (2𝑜 ·𝑜 𝑥))    &   𝑆 = (𝑥 ∈ On ↦ suc 𝑥)       ((𝐴𝑉 ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII)) → 𝐴 ∈ FinIII)

Theoremfin1a2lem8 8988* Lemma for fin1a2 8996. Split a III-infinite set in two pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.)
((𝐴𝑉 ∧ ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ FinIII ∨ (𝐴𝑥) ∈ FinIII)) → 𝐴 ∈ FinIII)

Theoremfin1a2lem9 8989* Lemma for fin1a2 8996. In a chain of finite sets, initial segments are finite. (Contributed by Stefan O'Rear, 8-Nov-2014.)
(( [] Or 𝑋𝑋 ⊆ Fin ∧ 𝐴 ∈ ω) → {𝑏𝑋𝑏𝐴} ∈ Fin)

Theoremfin1a2lem10 8990 Lemma for fin1a2 8996. A nonempty finite union of members of a chain is a member of the chain. (Contributed by Stefan O'Rear, 8-Nov-2014.)
((𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin ∧ [] Or 𝐴) → 𝐴𝐴)

Theoremfin1a2lem11 8991* Lemma for fin1a2 8996. (Contributed by Stefan O'Rear, 8-Nov-2014.)
(( [] Or 𝐴𝐴 ⊆ Fin) → ran (𝑏 ∈ ω ↦ {𝑐𝐴𝑐𝑏}) = (𝐴 ∪ {∅}))

Theoremfin1a2lem12 8992 Lemma for fin1a2 8996. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
(((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) → ¬ 𝐵 ∈ FinIII)

Theoremfin1a2lem13 8993 Lemma for fin1a2 8996. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
(((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) → ¬ (𝐵𝐶) ∈ FinII)

Theoremfin12 8994 Weak theorem which skips Ia but has a trivial proof, needed to prove fin1a2 8996. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
(𝐴 ∈ Fin → 𝐴 ∈ FinII)

Theoremfin1a2s 8995* An II-infinite set can have an I-infinite part broken off and remain II-infinite. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
((𝐴𝑉 ∧ ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴𝑥) ∈ FinII)) → 𝐴 ∈ FinII)

Theoremfin1a2 8996 Every Ia-finite set is II-finite. Theorem 1 of [Levy58], p. 3. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
(𝐴 ∈ FinIa𝐴 ∈ FinII)

2.6.13  Hereditarily size-limited sets without Choice

Theoremitunifval 8997* Function value of iterated unions. EDITORIAL: The iterated unions and order types of ordered sets are split out here because they could conceivably be independently useful. (Contributed by Stefan O'Rear, 11-Feb-2015.)
𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))       (𝐴𝑉 → (𝑈𝐴) = (rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω))

Theoremitunifn 8998* Functionality of the iterated union. (Contributed by Stefan O'Rear, 11-Feb-2015.)
𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))       (𝐴𝑉 → (𝑈𝐴) Fn ω)

Theoremituni0 8999* A zero-fold iterated union. (Contributed by Stefan O'Rear, 11-Feb-2015.)
𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))       (𝐴𝑉 → ((𝑈𝐴)‘∅) = 𝐴)

Theoremitunisuc 9000* Successor iterated union. (Contributed by Stefan O'Rear, 11-Feb-2015.)
𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))       ((𝑈𝐴)‘suc 𝐵) = ((𝑈𝐴)‘𝐵)

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