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Theorem List for Metamath Proof Explorer - 9501-9600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfin23lem32 9501* Lemma for fin23 9546. 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 9502* Lemma for fin23 9546. 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 9503* Lemma for fin23 9546. 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 9504* Lemma for fin23 9546. 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 9505* Lemma for fin23 9546. 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 9506* Lemma for fin23 9546. 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 9507* Lemma for fin23 9546. 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 9508* Lemma for fin23 9546. 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 9509* Lemma for fin23 9546. 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 9510* Lemma for isfin3-2 9524. 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 9511* Lemma for isfin3-2 9524. 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 9512* Lemma for isfin3-2 9524. 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 9513* Lemma for isfin3-2 9524. 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 9514* Lemma for isfin3-2 9524. 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 9515* Lemma for isfin3-2 9524. 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 9516* Lemma for isfin3-2 9524. 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 9517* Lemma for isfin3-2 9524. 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 9518* Lemma for isfin3-2 9524. 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 9519* 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 9520* Lemma for isfin3-2 9524. Remove hypotheses from isf32lem10 9519. (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 9521* Lemma for isfin3-2 9524. (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 9522 One half of isfin3-2 9524. (Contributed by Mario Carneiro, 3-Jun-2015.)
(𝐴 ∈ FinIII → ¬ ω ≼* 𝐴)

Theoremisf33lem 9523* Lemma for isfin3-3 9525. (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 9524 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 9525* 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 9526* Inference from isfin3-3 9525. (This is actually a bit stronger than isfin3-3 9525 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 9527* Lemma for compsscnv 9528. (Contributed by Mario Carneiro, 17-May-2015.)
((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦)))

Theoremcompsscnv 9528* Complementation on a power set lattice is an involution. (Contributed by Mario Carneiro, 17-May-2015.)
𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))       𝐹 = 𝐹

Theoremisf34lem1 9529* Lemma for isfin3-4 9539. (Contributed by Stefan O'Rear, 7-Nov-2014.)
𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))       ((𝐴𝑉𝑋𝐴) → (𝐹𝑋) = (𝐴𝑋))

Theoremisf34lem2 9530* Lemma for isfin3-4 9539. (Contributed by Stefan O'Rear, 7-Nov-2014.)
𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))       (𝐴𝑉𝐹:𝒫 𝐴⟶𝒫 𝐴)

Theoremcompssiso 9531* 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 9532* Lemma for isfin3-4 9539. (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 9533* 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 9534* Lemma for isfin3-4 9539. (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 9535* Lemma for isfin3-4 9539. (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 9536* Lemma for isfin3-4 9539. (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 9537* Lemma for isfin3-4 9539. (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 9538* Inference from isfin3-4 9539. (Contributed by Mario Carneiro, 17-May-2015.)
((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑥 ∈ ω (𝐺𝑥) ⊆ (𝐺‘suc 𝑥)) → ran 𝐺 ∈ ran 𝐺)

Theoremisfin3-4 9539* 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 9540 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 9541 Ia-finiteness is a cardinal property. (Contributed by Mario Carneiro, 18-May-2015.)
(𝐴𝐵 → (𝐴 ∈ FinIa𝐵 ∈ FinIa))

Theoremisfin1-2 9542 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 9543 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 9544 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 9545 Compact quantifier-free version of the standard definition df-fin 8245. (Contributed by Stefan O'Rear, 6-Jan-2015.)
Fin = ( ≈ “ ω)

Theoremfin23 9546 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 9475) 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 9603 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 9547 Every III-finite set is IV-finite. (Contributed by Stefan O'Rear, 30-Oct-2014.)
(𝐴 ∈ FinIII𝐴 ∈ FinIV)

Theoremisfin5-2 9548 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 9549 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 9550 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 9551 Every I-finite set is VII-finite. (Contributed by Mario Carneiro, 17-May-2015.)
(𝐴 ∈ Fin → 𝐴 ∈ FinVII)

Theoremfin67 9552 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 9553 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 9554 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 9555 Class form of isfin7-2 9553. (Contributed by Mario Carneiro, 17-May-2015.)
FinVII = (Fin ∪ (V ∖ dom card))

Theoremdfacfin7 9556 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 9557 Lemma for fin1a2 9572. (Contributed by Stefan O'Rear, 7-Nov-2014.)
𝑆 = (𝑥 ∈ On ↦ suc 𝑥)       (𝐴 ∈ On → (𝑆𝐴) = suc 𝐴)

Theoremfin1a2lem2 9558 Lemma for fin1a2 9572. (Contributed by Stefan O'Rear, 7-Nov-2014.)
𝑆 = (𝑥 ∈ On ↦ suc 𝑥)       𝑆:On–1-1→On

Theoremfin1a2lem3 9559 Lemma for fin1a2 9572. (Contributed by Stefan O'Rear, 7-Nov-2014.)
𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))       (𝐴 ∈ ω → (𝐸𝐴) = (2o ·o 𝐴))

Theoremfin1a2lem4 9560 Lemma for fin1a2 9572. (Contributed by Stefan O'Rear, 7-Nov-2014.)
𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))       𝐸:ω–1-1→ω

Theoremfin1a2lem5 9561 Lemma for fin1a2 9572. (Contributed by Stefan O'Rear, 7-Nov-2014.)
𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))       (𝐴 ∈ ω → (𝐴 ∈ ran 𝐸 ↔ ¬ suc 𝐴 ∈ ran 𝐸))

Theoremfin1a2lem6 9562 Lemma for fin1a2 9572. 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 9563* Lemma for fin1a2 9572. 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 9564* Lemma for fin1a2 9572. 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 9565* Lemma for fin1a2 9572. 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 9566 Lemma for fin1a2 9572. 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 9567* Lemma for fin1a2 9572. (Contributed by Stefan O'Rear, 8-Nov-2014.)
(( [] Or 𝐴𝐴 ⊆ Fin) → ran (𝑏 ∈ ω ↦ {𝑐𝐴𝑐𝑏}) = (𝐴 ∪ {∅}))

Theoremfin1a2lem12 9568 Lemma for fin1a2 9572. (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 9569 Lemma for fin1a2 9572. (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 9570 Weak theorem which skips Ia but has a trivial proof, needed to prove fin1a2 9572. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
(𝐴 ∈ Fin → 𝐴 ∈ FinII)

Theoremfin1a2s 9571* 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 9572 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)

2.6.14  Hereditarily size-limited sets without Choice

Theoremitunifval 9573* 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 9574* 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 9575* 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 9576* 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)

Theoremitunitc1 9577* Each union iterate is a member of the transitive closure. (Contributed by Stefan O'Rear, 11-Feb-2015.)
𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))       ((𝑈𝐴)‘𝐵) ⊆ (TC‘𝐴)

Theoremitunitc 9578* The union of all union iterates creates the transitive closure; compare trcl 8901. (Contributed by Stefan O'Rear, 11-Feb-2015.)
𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))       (TC‘𝐴) = ran (𝑈𝐴)

Theoremituniiun 9579* Unwrap an iterated union from the "other end". (Contributed by Stefan O'Rear, 11-Feb-2015.)
𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))       (𝐴𝑉 → ((𝑈𝐴)‘suc 𝐵) = 𝑎𝐴 ((𝑈𝑎)‘𝐵))

Theoremhsmexlem7 9580* Lemma for hsmex 9589. Properties of the recurrent sequence of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.)
𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)       (𝐻‘∅) = (har‘𝒫 𝑋)

Theoremhsmexlem8 9581* Lemma for hsmex 9589. Properties of the recurrent sequence of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.)
𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)       (𝑎 ∈ ω → (𝐻‘suc 𝑎) = (har‘𝒫 (𝑋 × (𝐻𝑎))))

Theoremhsmexlem9 9582* Lemma for hsmex 9589. Properties of the recurrent sequence of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.)
𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)       (𝑎 ∈ ω → (𝐻𝑎) ∈ On)

Theoremhsmexlem1 9583 Lemma for hsmex 9589. Bound the order type of a limited-cardinality set of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)
𝑂 = OrdIso( E , 𝐴)       ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂 ∈ (har‘𝒫 𝐵))

Theoremhsmexlem2 9584* Lemma for hsmex 9589. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 9732 but use of order types allows to canonically choose the sub-bijections, removing the choice requirement. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by AV, 18-Sep-2021.)
𝐹 = OrdIso( E , 𝐵)    &   𝐺 = OrdIso( E , 𝑎𝐴 𝐵)       ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐴 × 𝐶)))

Theoremhsmexlem3 9585* Lemma for hsmex 9589. Clear 𝐼 hypothesis and extend previous result by dominance. Note that this could be substantially strengthened, e.g., using the weak Hartogs function, but all we need here is that there be *some* dominating ordinal. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)
𝐹 = OrdIso( E , 𝐵)    &   𝐺 = OrdIso( E , 𝑎𝐴 𝐵)       (((𝐴* 𝐷𝐶 ∈ On) ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐷 × 𝐶)))

Theoremhsmexlem4 9586* Lemma for hsmex 9589. The core induction, establishing bounds on the order types of iterated unions of the initial set. (Contributed by Stefan O'Rear, 14-Feb-2015.)
𝑋 ∈ V    &   𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)    &   𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))    &   𝑆 = {𝑎 (𝑅1 “ On) ∣ ∀𝑏 ∈ (TC‘{𝑎})𝑏𝑋}    &   𝑂 = OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐)))       ((𝑐 ∈ ω ∧ 𝑑𝑆) → dom 𝑂 ∈ (𝐻𝑐))

Theoremhsmexlem5 9587* Lemma for hsmex 9589. Combining the above constraints, along with itunitc 9578 and tcrank 9044, gives an effective constraint on the rank of 𝑆. (Contributed by Stefan O'Rear, 14-Feb-2015.)
𝑋 ∈ V    &   𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)    &   𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))    &   𝑆 = {𝑎 (𝑅1 “ On) ∣ ∀𝑏 ∈ (TC‘{𝑎})𝑏𝑋}    &   𝑂 = OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐)))       (𝑑𝑆 → (rank‘𝑑) ∈ (har‘𝒫 (ω × ran 𝐻)))

Theoremhsmexlem6 9588* Lemma for hsmex 9589. (Contributed by Stefan O'Rear, 14-Feb-2015.)
𝑋 ∈ V    &   𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)    &   𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))    &   𝑆 = {𝑎 (𝑅1 “ On) ∣ ∀𝑏 ∈ (TC‘{𝑎})𝑏𝑋}    &   𝑂 = OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐)))       𝑆 ∈ V

Theoremhsmex 9589* The collection of hereditarily size-limited well-founded sets comprise a set. The proof is that of Randall Holmes at http://math.boisestate.edu/~holmes/holmes/hereditary.pdf, with modifications to use Hartogs' theorem instead of the weak variant (inconsequentially weakening some intermediate results), and making the well-foundedness condition explicit to avoid a direct dependence on ax-reg 8786. (Contributed by Stefan O'Rear, 14-Feb-2015.)
(𝑋𝑉 → {𝑠 (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑋} ∈ V)

Theoremhsmex2 9590* The set of hereditary size-limited sets, assuming ax-reg 8786. (Contributed by Stefan O'Rear, 11-Feb-2015.)
(𝑋𝑉 → {𝑠 ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑋} ∈ V)

Theoremhsmex3 9591* The set of hereditary size-limited sets, assuming ax-reg 8786, using strict comparison (an easy corollary by separation). (Contributed by Stefan O'Rear, 11-Feb-2015.)
(𝑋𝑉 → {𝑠 ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥𝑋} ∈ V)

PART 3  ZFC (ZERMELO-FRAENKEL WITH CHOICE) SET THEORY

In this section we add the Axiom of Choice ax-ac 9616, as well as weaker forms such as the axiom of countable choice ax-cc 9592 and dependent choice ax-dc 9603. We introduce these weaker forms so that theorems that do not need the full power of the axiom of choice, but need more than simple ZF, can use these intermediate axioms instead.

The combination of the Zermelo-Fraenkel axioms and the axiom of choice is often abbreviated as ZFC. The axiom of choice is widely accepted, and ZFC is the most commonly-accepted fundamental set of axioms for mathematics.

However, there have been and still are some lingering controversies about the Axiom of Choice. The axiom of choice does not satisfy those who wish to have a constructive proof (e.g., it will not satisfy intuitionistic logic). Thus, we make it easy to identify which proofs depend on the axiom of choice or its weaker forms.

3.1  ZFC Set Theory - add Countable Choice and Dependent Choice

3.1.1  Introduce the Axiom of Countable Choice

Axiomax-cc 9592* The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 9634, but is weak enough that it can be proven using DC (see axcc 9615). It is, however, strictly stronger than ZF and cannot be proven in ZF. It states that any countable collection of nonempty sets must have a choice function. (Contributed by Mario Carneiro, 9-Feb-2013.)
(𝑥 ≈ ω → ∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))

Theoremaxcc2lem 9593* Lemma for axcc2 9594. (Contributed by Mario Carneiro, 8-Feb-2013.)
𝐾 = (𝑛 ∈ ω ↦ if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))    &   𝐴 = (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛)))    &   𝐺 = (𝑛 ∈ ω ↦ (2nd ‘(𝑓‘(𝐴𝑛))))       𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛)))

Theoremaxcc2 9594* A possibly more useful version of ax-cc using sequences instead of countable sets. The Axiom of Infinity is needed to prove this, and indeed this implies the Axiom of Infinity. (Contributed by Mario Carneiro, 8-Feb-2013.)
𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛)))

Theoremaxcc3 9595* A possibly more useful version of ax-cc 9592 using sequences 𝐹(𝑛) instead of countable sets. The Axiom of Infinity is needed to prove this, and indeed this implies the Axiom of Infinity. (Contributed by Mario Carneiro, 8-Feb-2013.) (Revised by Mario Carneiro, 26-Dec-2014.)
𝐹 ∈ V    &   𝑁 ≈ ω       𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹))

Theoremaxcc4 9596* A version of axcc3 9595 that uses wffs instead of classes. (Contributed by Mario Carneiro, 7-Apr-2013.)
𝐴 ∈ V    &   𝑁 ≈ ω    &   (𝑥 = (𝑓𝑛) → (𝜑𝜓))       (∀𝑛𝑁𝑥𝐴 𝜑 → ∃𝑓(𝑓:𝑁𝐴 ∧ ∀𝑛𝑁 𝜓))

Theoremacncc 9597 An ax-cc 9592 equivalent: every set has choice sets of length ω. (Contributed by Mario Carneiro, 31-Aug-2015.)
AC ω = V

Theoremaxcc4dom 9598* Relax the constraint on axcc4 9596 to dominance instead of equinumerosity. (Contributed by Mario Carneiro, 18-Jan-2014.)
𝐴 ∈ V    &   (𝑥 = (𝑓𝑛) → (𝜑𝜓))       ((𝑁 ≼ ω ∧ ∀𝑛𝑁𝑥𝐴 𝜑) → ∃𝑓(𝑓:𝑁𝐴 ∧ ∀𝑛𝑁 𝜓))

Theoremdomtriomlem 9599* Lemma for domtriom 9600. (Contributed by Mario Carneiro, 9-Feb-2013.)
𝐴 ∈ V    &   𝐵 = {𝑦 ∣ (𝑦𝐴𝑦 ≈ 𝒫 𝑛)}    &   𝐶 = (𝑛 ∈ ω ↦ ((𝑏𝑛) ∖ 𝑘𝑛 (𝑏𝑘)))       𝐴 ∈ Fin → ω ≼ 𝐴)

Theoremdomtriom 9600 Trichotomy of equinumerosity for ω, proven using countable choice. Equivalently, all Dedekind-finite sets (as in isfin4-2 9471) are finite in the usual sense and conversely. (Contributed by Mario Carneiro, 9-Feb-2013.)
𝐴 ∈ V       (ω ≼ 𝐴 ↔ ¬ 𝐴 ≺ ω)

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