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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | infpssrlem4 10301 | Lemma for infpssr 10303. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
⊢ (𝜑 → 𝐵 ⊆ 𝐴) & ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝐴) & ⊢ (𝜑 → 𝐶 ∈ (𝐴 ∖ 𝐵)) & ⊢ 𝐺 = (rec(◡𝐹, 𝐶) ↾ ω) ⇒ ⊢ ((𝜑 ∧ 𝑀 ∈ ω ∧ 𝑁 ∈ 𝑀) → (𝐺‘𝑀) ≠ (𝐺‘𝑁)) | ||
Theorem | infpssrlem5 10302 | Lemma for infpssr 10303. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
⊢ (𝜑 → 𝐵 ⊆ 𝐴) & ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝐴) & ⊢ (𝜑 → 𝐶 ∈ (𝐴 ∖ 𝐵)) & ⊢ 𝐺 = (rec(◡𝐹, 𝐶) ↾ ω) ⇒ ⊢ (𝜑 → (𝐴 ∈ 𝑉 → ω ≼ 𝐴)) | ||
Theorem | infpssr 10303 | Dedekind infinity implies existence of a denumerable subset: take a single point witnessing the proper subset relation and iterate the embedding. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.) |
⊢ ((𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴) → ω ≼ 𝐴) | ||
Theorem | fin4en1 10304 | Dedekind finite is a cardinal property. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.) |
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ FinIV → 𝐵 ∈ FinIV)) | ||
Theorem | ssfin4 10305 | Dedekind finite sets have Dedekind finite subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 6-May-2015.) (Revised by Mario Carneiro, 16-May-2015.) |
⊢ ((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ FinIV) | ||
Theorem | domfin4 10306 | A set dominated by a Dedekind finite set is Dedekind finite. (Contributed by Mario Carneiro, 16-May-2015.) |
⊢ ((𝐴 ∈ FinIV ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ FinIV) | ||
Theorem | ominf4 10307 | ω is Dedekind infinite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Proof shortened by Mario Carneiro, 16-May-2015.) |
⊢ ¬ ω ∈ FinIV | ||
Theorem | infpssALT 10308* | Alternate proof of infpss 10212, shorter but requiring Replacement (ax-rep 5286). (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (ω ≼ 𝐴 → ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴)) | ||
Theorem | isfin4-2 10309 | Alternate definition of IV-finite sets: they lack a denumerable subset. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝐴)) | ||
Theorem | isfin4p1 10310 | Alternate definition of IV-finite sets: they are strictly dominated by their successors. (Thus, the proper subset referred to in isfin4 10292 can be assumed to be only a singleton smaller than the original.) (Contributed by Mario Carneiro, 18-May-2015.) |
⊢ (𝐴 ∈ FinIV ↔ 𝐴 ≺ (𝐴 ⊔ 1o)) | ||
Theorem | fin23lem7 10311* | Lemma for isfin2-2 10314. The componentwise complement of a nonempty collection of sets is nonempty. (Contributed by Stefan O'Rear, 31-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴 ∧ 𝐵 ≠ ∅) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑥) ∈ 𝐵} ≠ ∅) | ||
Theorem | fin23lem11 10312* | Lemma for isfin2-2 10314. (Contributed by Stefan O'Rear, 31-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.) |
⊢ (𝑧 = (𝐴 ∖ 𝑥) → (𝜓 ↔ 𝜒)) & ⊢ (𝑤 = (𝐴 ∖ 𝑣) → (𝜑 ↔ 𝜃)) & ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑣 ⊆ 𝐴) → (𝜒 ↔ 𝜃)) ⇒ ⊢ (𝐵 ⊆ 𝒫 𝐴 → (∃𝑥 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵}∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑐) ∈ 𝐵} ¬ 𝜑 → ∃𝑧 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ¬ 𝜓)) | ||
Theorem | fin2i2 10313 | A II-finite set contains minimal elements for every nonempty chain. (Contributed by Mario Carneiro, 16-May-2015.) |
⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [⊊] Or 𝐵)) → ∩ 𝐵 ∈ 𝐵) | ||
Theorem | isfin2-2 10314* | FinII expressed in terms of minimal elements. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Proof shortened by Mario Carneiro, 16-May-2015.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinII ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∩ 𝑦 ∈ 𝑦))) | ||
Theorem | ssfin2 10315 | A subset of a II-finite set is II-finite. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 16-May-2015.) |
⊢ ((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ FinII) | ||
Theorem | enfin2i 10316 | II-finiteness is a cardinal property. (Contributed by Mario Carneiro, 18-May-2015.) |
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ FinII → 𝐵 ∈ FinII)) | ||
Theorem | fin23lem24 10317 | Lemma for fin23 10384. In a class of ordinals, each element is fully identified by those of its predecessors which also belong to the class. (Contributed by Stefan O'Rear, 1-Nov-2014.) |
⊢ (((Ord 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → ((𝐶 ∩ 𝐵) = (𝐷 ∩ 𝐵) ↔ 𝐶 = 𝐷)) | ||
Theorem | fincssdom 10318 | In a chain of finite sets, dominance and subset coincide. (Contributed by Stefan O'Rear, 8-Nov-2014.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) → (𝐴 ≼ 𝐵 ↔ 𝐴 ⊆ 𝐵)) | ||
Theorem | fin23lem25 10319 | Lemma for fin23 10384. In a chain of finite sets, equinumerosity is equivalent to equality. (Contributed by Stefan O'Rear, 1-Nov-2014.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) → (𝐴 ≈ 𝐵 ↔ 𝐴 = 𝐵)) | ||
Theorem | fin23lem26 10320* | Lemma for fin23lem22 10322. (Contributed by Stefan O'Rear, 1-Nov-2014.) |
⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖) | ||
Theorem | fin23lem23 10321* | Lemma for fin23lem22 10322. (Contributed by Stefan O'Rear, 1-Nov-2014.) |
⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃!𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖) | ||
Theorem | fin23lem22 10322* | Lemma for fin23 10384 but could be used elsewhere if we find a good name for it. Explicit construction of a bijection (actually an isomorphism, see fin23lem27 10323) between an infinite subset of ω and ω itself. (Contributed by Stefan O'Rear, 1-Nov-2014.) |
⊢ 𝐶 = (𝑖 ∈ ω ↦ (℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖)) ⇒ ⊢ ((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) → 𝐶:ω–1-1-onto→𝑆) | ||
Theorem | fin23lem27 10323* | The mapping constructed in fin23lem22 10322 is in fact an isomorphism. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
⊢ 𝐶 = (𝑖 ∈ ω ↦ (℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖)) ⇒ ⊢ ((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) → 𝐶 Isom E , E (ω, 𝑆)) | ||
Theorem | isfin3ds 10324* | Property of a III-finite set (descending sequence version). (Contributed by Mario Carneiro, 16-May-2015.) |
⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎‘𝑏) → ∩ ran 𝑎 ∈ ran 𝑎)} ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐹 ↔ ∀𝑓 ∈ (𝒫 𝐴 ↑m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥) → ∩ ran 𝑓 ∈ ran 𝑓))) | ||
Theorem | ssfin3ds 10325* | A subset of a III-finite set is III-finite. (Contributed by Stefan O'Rear, 4-Nov-2014.) |
⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎‘𝑏) → ∩ ran 𝑎 ∈ ran 𝑎)} ⇒ ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ 𝐹) | ||
Theorem | fin23lem12 10326* |
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(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡) ⇒ ⊢ (𝐴 ∈ ω → (𝑈‘suc 𝐴) = if(((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅, (𝑈‘𝐴), ((𝑡‘𝐴) ∩ (𝑈‘𝐴)))) | ||
Theorem | fin23lem13 10327* | Lemma for fin23 10384. Each step of 𝑈 is a decrease. (Contributed by Stefan O'Rear, 1-Nov-2014.) |
⊢ 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡) ⇒ ⊢ (𝐴 ∈ ω → (𝑈‘suc 𝐴) ⊆ (𝑈‘𝐴)) | ||
Theorem | fin23lem14 10328* | Lemma for fin23 10384. 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(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡) ⇒ ⊢ ((𝐴 ∈ ω ∧ ∪ ran 𝑡 ≠ ∅) → (𝑈‘𝐴) ≠ ∅) | ||
Theorem | fin23lem15 10329* | Lemma for fin23 10384. 𝑈 is a monotone function. (Contributed by Stefan O'Rear, 1-Nov-2014.) |
⊢ 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡) ⇒ ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝐴) → (𝑈‘𝐴) ⊆ (𝑈‘𝐵)) | ||
Theorem | fin23lem16 10330* | Lemma for fin23 10384. 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(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡) ⇒ ⊢ ∪ ran 𝑈 = ∪ ran 𝑡 | ||
Theorem | fin23lem19 10331* | Lemma for fin23 10384. 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(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡) ⇒ ⊢ (𝐴 ∈ ω → ((𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴) ∨ ((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ∅)) | ||
Theorem | fin23lem20 10332* | Lemma for fin23 10384. 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(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡) ⇒ ⊢ (𝐴 ∈ ω → (∩ ran 𝑈 ⊆ (𝑡‘𝐴) ∨ (∩ ran 𝑈 ∩ (𝑡‘𝐴)) = ∅)) | ||
Theorem | fin23lem17 10333* | Lemma for fin23 10384. 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(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡) & ⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)} ⇒ ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → ∩ ran 𝑈 ∈ ran 𝑈) | ||
Theorem | fin23lem21 10334* | Lemma for fin23 10384. 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(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡) & ⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)} ⇒ ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → ∩ ran 𝑈 ≠ ∅) | ||
Theorem | fin23lem28 10335* | Lemma for fin23 10384. 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(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡) & ⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)} & ⊢ 𝑃 = {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)} & ⊢ 𝑄 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ 𝑃 (𝑥 ∩ 𝑃) ≈ 𝑤)) & ⊢ 𝑅 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤)) & ⊢ 𝑍 = if(𝑃 ∈ Fin, (𝑡 ∘ 𝑅), ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄)) ⇒ ⊢ (𝑡:ω–1-1→V → 𝑍:ω–1-1→V) | ||
Theorem | fin23lem29 10336* | Lemma for fin23 10384. 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(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡) & ⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)} & ⊢ 𝑃 = {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)} & ⊢ 𝑄 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ 𝑃 (𝑥 ∩ 𝑃) ≈ 𝑤)) & ⊢ 𝑅 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤)) & ⊢ 𝑍 = if(𝑃 ∈ Fin, (𝑡 ∘ 𝑅), ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄)) ⇒ ⊢ ∪ ran 𝑍 ⊆ ∪ ran 𝑡 | ||
Theorem | fin23lem30 10337* | Lemma for fin23 10384. The residual is disjoint from the common set. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
⊢ 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡) & ⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)} & ⊢ 𝑃 = {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)} & ⊢ 𝑄 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ 𝑃 (𝑥 ∩ 𝑃) ≈ 𝑤)) & ⊢ 𝑅 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤)) & ⊢ 𝑍 = if(𝑃 ∈ Fin, (𝑡 ∘ 𝑅), ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄)) ⇒ ⊢ (Fun 𝑡 → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅) | ||
Theorem | fin23lem31 10338* | Lemma for fin23 10384. 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(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡) & ⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)} & ⊢ 𝑃 = {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)} & ⊢ 𝑄 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ 𝑃 (𝑥 ∩ 𝑃) ≈ 𝑤)) & ⊢ 𝑅 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤)) & ⊢ 𝑍 = if(𝑃 ∈ Fin, (𝑡 ∘ 𝑅), ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄)) ⇒ ⊢ ((𝑡:ω–1-1→𝑉 ∧ 𝐺 ∈ 𝐹 ∧ ∪ ran 𝑡 ⊆ 𝐺) → ∪ ran 𝑍 ⊊ ∪ ran 𝑡) | ||
Theorem | fin23lem32 10339* | Lemma for fin23 10384. 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(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡) & ⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)} & ⊢ 𝑃 = {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)} & ⊢ 𝑄 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ 𝑃 (𝑥 ∩ 𝑃) ≈ 𝑤)) & ⊢ 𝑅 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤)) & ⊢ 𝑍 = if(𝑃 ∈ Fin, (𝑡 ∘ 𝑅), ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄)) ⇒ ⊢ (𝐺 ∈ 𝐹 → ∃𝑓∀𝑏((𝑏:ω–1-1→V ∧ ∪ ran 𝑏 ⊆ 𝐺) → ((𝑓‘𝑏):ω–1-1→V ∧ ∪ ran (𝑓‘𝑏) ⊊ ∪ ran 𝑏))) | ||
Theorem | fin23lem33 10340* | Lemma for fin23 10384. Discharge hypotheses. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)} ⇒ ⊢ (𝐺 ∈ 𝐹 → ∃𝑓∀𝑏((𝑏:ω–1-1→V ∧ ∪ ran 𝑏 ⊆ 𝐺) → ((𝑓‘𝑏):ω–1-1→V ∧ ∪ ran (𝑓‘𝑏) ⊊ ∪ ran 𝑏))) | ||
Theorem | fin23lem34 10341* | Lemma for fin23 10384. Establish induction invariants on 𝑌 which parameterizes our contradictory chain of subsets. In this section, ℎ 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 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)} & ⊢ (𝜑 → ℎ:ω–1-1→V) & ⊢ (𝜑 → ∪ ran ℎ ⊆ 𝐺) & ⊢ (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ∪ ran 𝑗 ⊆ 𝐺) → ((𝑖‘𝑗):ω–1-1→V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗))) & ⊢ 𝑌 = (rec(𝑖, ℎ) ↾ ω) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ((𝑌‘𝐴):ω–1-1→V ∧ ∪ ran (𝑌‘𝐴) ⊆ 𝐺)) | ||
Theorem | fin23lem35 10342* | Lemma for fin23 10384. Strict order property of 𝑌. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)} & ⊢ (𝜑 → ℎ:ω–1-1→V) & ⊢ (𝜑 → ∪ ran ℎ ⊆ 𝐺) & ⊢ (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ∪ ran 𝑗 ⊆ 𝐺) → ((𝑖‘𝑗):ω–1-1→V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗))) & ⊢ 𝑌 = (rec(𝑖, ℎ) ↾ ω) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ∪ ran (𝑌‘suc 𝐴) ⊊ ∪ ran (𝑌‘𝐴)) | ||
Theorem | fin23lem36 10343* | Lemma for fin23 10384. Weak order property of 𝑌. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)} & ⊢ (𝜑 → ℎ:ω–1-1→V) & ⊢ (𝜑 → ∪ ran ℎ ⊆ 𝐺) & ⊢ (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ∪ ran 𝑗 ⊆ 𝐺) → ((𝑖‘𝑗):ω–1-1→V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗))) & ⊢ 𝑌 = (rec(𝑖, ℎ) ↾ ω) ⇒ ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵 ⊆ 𝐴 ∧ 𝜑)) → ∪ ran (𝑌‘𝐴) ⊆ ∪ ran (𝑌‘𝐵)) | ||
Theorem | fin23lem38 10344* | Lemma for fin23 10384. 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 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)} & ⊢ (𝜑 → ℎ:ω–1-1→V) & ⊢ (𝜑 → ∪ ran ℎ ⊆ 𝐺) & ⊢ (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ∪ ran 𝑗 ⊆ 𝐺) → ((𝑖‘𝑗):ω–1-1→V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗))) & ⊢ 𝑌 = (rec(𝑖, ℎ) ↾ ω) ⇒ ⊢ (𝜑 → ¬ ∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ∈ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏))) | ||
Theorem | fin23lem39 10345* | Lemma for fin23 10384. 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 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)} & ⊢ (𝜑 → ℎ:ω–1-1→V) & ⊢ (𝜑 → ∪ ran ℎ ⊆ 𝐺) & ⊢ (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ∪ ran 𝑗 ⊆ 𝐺) → ((𝑖‘𝑗):ω–1-1→V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗))) & ⊢ 𝑌 = (rec(𝑖, ℎ) ↾ ω) ⇒ ⊢ (𝜑 → ¬ 𝐺 ∈ 𝐹) | ||
Theorem | fin23lem40 10346* | Lemma for fin23 10384. FinII sets satisfy the descending chain condition. (Contributed by Stefan O'Rear, 3-Nov-2014.) |
⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)} ⇒ ⊢ (𝐴 ∈ FinII → 𝐴 ∈ 𝐹) | ||
Theorem | fin23lem41 10347* | Lemma for fin23 10384. A set which satisfies the descending sequence condition must be III-finite. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)} ⇒ ⊢ (𝐴 ∈ 𝐹 → 𝐴 ∈ FinIII) | ||
Theorem | isf32lem1 10348* | Lemma for isfin3-2 10362. Derive weak ordering property. (Contributed by Stefan O'Rear, 5-Nov-2014.) |
⊢ (𝜑 → 𝐹:ω⟶𝒫 𝐺) & ⊢ (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥)) & ⊢ (𝜑 → ¬ ∩ ran 𝐹 ∈ ran 𝐹) ⇒ ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵 ⊆ 𝐴 ∧ 𝜑)) → (𝐹‘𝐴) ⊆ (𝐹‘𝐵)) | ||
Theorem | isf32lem2 10349* | Lemma for isfin3-2 10362. Non-minimum implies that there is always another decrease. (Contributed by Stefan O'Rear, 5-Nov-2014.) |
⊢ (𝜑 → 𝐹:ω⟶𝒫 𝐺) & ⊢ (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥)) & ⊢ (𝜑 → ¬ ∩ ran 𝐹 ∈ ran 𝐹) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ∃𝑎 ∈ ω (𝐴 ∈ 𝑎 ∧ (𝐹‘suc 𝑎) ⊊ (𝐹‘𝑎))) | ||
Theorem | isf32lem3 10350* | Lemma for isfin3-2 10362. Being a chain, difference sets are disjoint (one case). (Contributed by Stefan O'Rear, 5-Nov-2014.) |
⊢ (𝜑 → 𝐹:ω⟶𝒫 𝐺) & ⊢ (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥)) & ⊢ (𝜑 → ¬ ∩ ran 𝐹 ∈ ran 𝐹) ⇒ ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵 ∈ 𝐴 ∧ 𝜑)) → (((𝐹‘𝐴) ∖ (𝐹‘suc 𝐴)) ∩ ((𝐹‘𝐵) ∖ (𝐹‘suc 𝐵))) = ∅) | ||
Theorem | isf32lem4 10351* | Lemma for isfin3-2 10362. Being a chain, difference sets are disjoint. (Contributed by Stefan O'Rear, 5-Nov-2014.) |
⊢ (𝜑 → 𝐹:ω⟶𝒫 𝐺) & ⊢ (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥)) & ⊢ (𝜑 → ¬ ∩ ran 𝐹 ∈ ran 𝐹) ⇒ ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (((𝐹‘𝐴) ∖ (𝐹‘suc 𝐴)) ∩ ((𝐹‘𝐵) ∖ (𝐹‘suc 𝐵))) = ∅) | ||
Theorem | isf32lem5 10352* | Lemma for isfin3-2 10362. There are infinite decrease points. (Contributed by Stefan O'Rear, 5-Nov-2014.) |
⊢ (𝜑 → 𝐹:ω⟶𝒫 𝐺) & ⊢ (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥)) & ⊢ (𝜑 → ¬ ∩ ran 𝐹 ∈ ran 𝐹) & ⊢ 𝑆 = {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)} ⇒ ⊢ (𝜑 → ¬ 𝑆 ∈ Fin) | ||
Theorem | isf32lem6 10353* | Lemma for isfin3-2 10362. Each K value is nonempty. (Contributed by Stefan O'Rear, 5-Nov-2014.) |
⊢ (𝜑 → 𝐹:ω⟶𝒫 𝐺) & ⊢ (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥)) & ⊢ (𝜑 → ¬ ∩ ran 𝐹 ∈ ran 𝐹) & ⊢ 𝑆 = {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)} & ⊢ 𝐽 = (𝑢 ∈ ω ↦ (℩𝑣 ∈ 𝑆 (𝑣 ∩ 𝑆) ≈ 𝑢)) & ⊢ 𝐾 = ((𝑤 ∈ 𝑆 ↦ ((𝐹‘𝑤) ∖ (𝐹‘suc 𝑤))) ∘ 𝐽) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝐾‘𝐴) ≠ ∅) | ||
Theorem | isf32lem7 10354* | Lemma for isfin3-2 10362. Different K values are disjoint. (Contributed by Stefan O'Rear, 5-Nov-2014.) |
⊢ (𝜑 → 𝐹:ω⟶𝒫 𝐺) & ⊢ (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥)) & ⊢ (𝜑 → ¬ ∩ ran 𝐹 ∈ ran 𝐹) & ⊢ 𝑆 = {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)} & ⊢ 𝐽 = (𝑢 ∈ ω ↦ (℩𝑣 ∈ 𝑆 (𝑣 ∩ 𝑆) ≈ 𝑢)) & ⊢ 𝐾 = ((𝑤 ∈ 𝑆 ↦ ((𝐹‘𝑤) ∖ (𝐹‘suc 𝑤))) ∘ 𝐽) ⇒ ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝐾‘𝐴) ∩ (𝐾‘𝐵)) = ∅) | ||
Theorem | isf32lem8 10355* | Lemma for isfin3-2 10362. K sets are subsets of the base. (Contributed by Stefan O'Rear, 6-Nov-2014.) |
⊢ (𝜑 → 𝐹:ω⟶𝒫 𝐺) & ⊢ (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥)) & ⊢ (𝜑 → ¬ ∩ ran 𝐹 ∈ ran 𝐹) & ⊢ 𝑆 = {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)} & ⊢ 𝐽 = (𝑢 ∈ ω ↦ (℩𝑣 ∈ 𝑆 (𝑣 ∩ 𝑆) ≈ 𝑢)) & ⊢ 𝐾 = ((𝑤 ∈ 𝑆 ↦ ((𝐹‘𝑤) ∖ (𝐹‘suc 𝑤))) ∘ 𝐽) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝐾‘𝐴) ⊆ 𝐺) | ||
Theorem | isf32lem9 10356* | Lemma for isfin3-2 10362. Construction of the onto function. (Contributed by Stefan O'Rear, 5-Nov-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) |
⊢ (𝜑 → 𝐹:ω⟶𝒫 𝐺) & ⊢ (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥)) & ⊢ (𝜑 → ¬ ∩ ran 𝐹 ∈ ran 𝐹) & ⊢ 𝑆 = {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)} & ⊢ 𝐽 = (𝑢 ∈ ω ↦ (℩𝑣 ∈ 𝑆 (𝑣 ∩ 𝑆) ≈ 𝑢)) & ⊢ 𝐾 = ((𝑤 ∈ 𝑆 ↦ ((𝐹‘𝑤) ∖ (𝐹‘suc 𝑤))) ∘ 𝐽) & ⊢ 𝐿 = (𝑡 ∈ 𝐺 ↦ (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠)))) ⇒ ⊢ (𝜑 → 𝐿:𝐺–onto→ω) | ||
Theorem | isf32lem10 10357* | 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.) |
⊢ (𝜑 → 𝐹:ω⟶𝒫 𝐺) & ⊢ (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥)) & ⊢ (𝜑 → ¬ ∩ ran 𝐹 ∈ ran 𝐹) & ⊢ 𝑆 = {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)} & ⊢ 𝐽 = (𝑢 ∈ ω ↦ (℩𝑣 ∈ 𝑆 (𝑣 ∩ 𝑆) ≈ 𝑢)) & ⊢ 𝐾 = ((𝑤 ∈ 𝑆 ↦ ((𝐹‘𝑤) ∖ (𝐹‘suc 𝑤))) ∘ 𝐽) & ⊢ 𝐿 = (𝑡 ∈ 𝐺 ↦ (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠)))) ⇒ ⊢ (𝜑 → (𝐺 ∈ 𝑉 → ω ≼* 𝐺)) | ||
Theorem | isf32lem11 10358* | Lemma for isfin3-2 10362. Remove hypotheses from isf32lem10 10357. (Contributed by Stefan O'Rear, 17-May-2015.) |
⊢ ((𝐺 ∈ 𝑉 ∧ (𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏) ∧ ¬ ∩ ran 𝐹 ∈ ran 𝐹)) → ω ≼* 𝐺) | ||
Theorem | isf32lem12 10359* | Lemma for isfin3-2 10362. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)} ⇒ ⊢ (𝐺 ∈ 𝑉 → (¬ ω ≼* 𝐺 → 𝐺 ∈ 𝐹)) | ||
Theorem | isfin32i 10360 | One half of isfin3-2 10362. (Contributed by Mario Carneiro, 3-Jun-2015.) |
⊢ (𝐴 ∈ FinIII → ¬ ω ≼* 𝐴) | ||
Theorem | isf33lem 10361* | Lemma for isfin3-3 10363. (Contributed by Stefan O'Rear, 17-May-2015.) |
⊢ FinIII = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)} | ||
Theorem | isfin3-2 10362 | 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.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIII ↔ ¬ ω ≼* 𝐴)) | ||
Theorem | isfin3-3 10363* | Weakly Dedekind-infinite sets are exactly those with an ω-indexed descending chain of subsets. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIII ↔ ∀𝑓 ∈ (𝒫 𝐴 ↑m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥) → ∩ ran 𝑓 ∈ ran 𝑓))) | ||
Theorem | fin33i 10364* | Inference from isfin3-3 10363. (This is actually a bit stronger than isfin3-3 10363 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 ∈ FinIII ∧ 𝐹:ω⟶𝒫 𝐴 ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥)) → ∩ ran 𝐹 ∈ ran 𝐹) | ||
Theorem | compsscnvlem 10365* | Lemma for compsscnv 10366. (Contributed by Mario Carneiro, 17-May-2015.) |
⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))) | ||
Theorem | compsscnv 10366* | Complementation on a power set lattice is an involution. (Contributed by Mario Carneiro, 17-May-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) ⇒ ⊢ ◡𝐹 = 𝐹 | ||
Theorem | isf34lem1 10367* | Lemma for isfin3-4 10377. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝐹‘𝑋) = (𝐴 ∖ 𝑋)) | ||
Theorem | isf34lem2 10368* | Lemma for isfin3-4 10377. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐹:𝒫 𝐴⟶𝒫 𝐴) | ||
Theorem | compssiso 10369* | 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 ↦ (𝐴 ∖ 𝑥)) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐹 Isom [⊊] , ◡ [⊊] (𝒫 𝐴, 𝒫 𝐴)) | ||
Theorem | isf34lem3 10370* | Lemma for isfin3-4 10377. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹 “ 𝑋)) = 𝑋) | ||
Theorem | compss 10371* | 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 ↦ (𝐴 ∖ 𝑥)) ⇒ ⊢ (𝐹 “ 𝐺) = {𝑦 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑦) ∈ 𝐺} | ||
Theorem | isf34lem4 10372* | Lemma for isfin3-4 10377. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹‘∪ 𝑋) = ∩ (𝐹 “ 𝑋)) | ||
Theorem | isf34lem5 10373* | Lemma for isfin3-4 10377. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹‘∩ 𝑋) = ∪ (𝐹 “ 𝑋)) | ||
Theorem | isf34lem7 10374* | Lemma for isfin3-4 10377. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) ⇒ ⊢ ((𝐴 ∈ FinIII ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → ∪ ran 𝐺 ∈ ran 𝐺) | ||
Theorem | isf34lem6 10375* | Lemma for isfin3-4 10377. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIII ↔ ∀𝑓 ∈ (𝒫 𝐴 ↑m ω)(∀𝑦 ∈ ω (𝑓‘𝑦) ⊆ (𝑓‘suc 𝑦) → ∪ ran 𝑓 ∈ ran 𝑓))) | ||
Theorem | fin34i 10376* | Inference from isfin3-4 10377. (Contributed by Mario Carneiro, 17-May-2015.) |
⊢ ((𝐴 ∈ FinIII ∧ 𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑥 ∈ ω (𝐺‘𝑥) ⊆ (𝐺‘suc 𝑥)) → ∪ ran 𝐺 ∈ ran 𝐺) | ||
Theorem | isfin3-4 10377* | 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.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIII ↔ ∀𝑓 ∈ (𝒫 𝐴 ↑m ω)(∀𝑥 ∈ ω (𝑓‘𝑥) ⊆ (𝑓‘suc 𝑥) → ∪ ran 𝑓 ∈ ran 𝑓))) | ||
Theorem | fin11a 10378 | 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) | ||
Theorem | enfin1ai 10379 | Ia-finiteness is a cardinal property. (Contributed by Mario Carneiro, 18-May-2015.) |
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ FinIa → 𝐵 ∈ FinIa)) | ||
Theorem | isfin1-2 10380 | 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) | ||
Theorem | isfin1-3 10381 | 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.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin ↔ ◡ [⊊] Fr 𝒫 𝐴)) | ||
Theorem | isfin1-4 10382 | 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.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin ↔ [⊊] Fr 𝒫 𝐴)) | ||
Theorem | dffin1-5 10383 | Compact quantifier-free version of the standard definition df-fin 8943. (Contributed by Stefan O'Rear, 6-Jan-2015.) |
⊢ Fin = ( ≈ “ ω) | ||
Theorem | fin23 10384 |
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 (𝑈‘𝑇) 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 10313) 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 10441 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 ∈ FinII → 𝐴 ∈ FinIII) | ||
Theorem | fin34 10385 | Every III-finite set is IV-finite. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
⊢ (𝐴 ∈ FinIII → 𝐴 ∈ FinIV) | ||
Theorem | isfin5-2 10386 | 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.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinV ↔ ¬ (𝐴 ≠ ∅ ∧ 𝐴 ≈ (𝐴 ⊔ 𝐴)))) | ||
Theorem | fin45 10387 | 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 ∈ FinIV → 𝐴 ∈ FinV) | ||
Theorem | fin56 10388 | 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 ∈ FinV → 𝐴 ∈ FinVI) | ||
Theorem | fin17 10389 | Every I-finite set is VII-finite. (Contributed by Mario Carneiro, 17-May-2015.) |
⊢ (𝐴 ∈ Fin → 𝐴 ∈ FinVII) | ||
Theorem | fin67 10390 | Every VI-finite set is VII-finite. (Contributed by Stefan O'Rear, 29-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
⊢ (𝐴 ∈ FinVI → 𝐴 ∈ FinVII) | ||
Theorem | isfin7-2 10391 | A set is VII-finite iff it is non-well-orderable or finite. (Contributed by Mario Carneiro, 17-May-2015.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinVII ↔ (𝐴 ∈ dom card → 𝐴 ∈ Fin))) | ||
Theorem | fin71num 10392 | 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 ∈ FinVII ↔ 𝐴 ∈ Fin)) | ||
Theorem | dffin7-2 10393 | Class form of isfin7-2 10391. (Contributed by Mario Carneiro, 17-May-2015.) |
⊢ FinVII = (Fin ∪ (V ∖ dom card)) | ||
Theorem | dfacfin7 10394 | 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) | ||
Theorem | fin1a2lem1 10395 | Lemma for fin1a2 10410. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
⊢ 𝑆 = (𝑥 ∈ On ↦ suc 𝑥) ⇒ ⊢ (𝐴 ∈ On → (𝑆‘𝐴) = suc 𝐴) | ||
Theorem | fin1a2lem2 10396 | Lemma for fin1a2 10410. The successor operation on the ordinal numbers is injective or one-to-one. Lemma 1.17 of [Schloeder] p. 2. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
⊢ 𝑆 = (𝑥 ∈ On ↦ suc 𝑥) ⇒ ⊢ 𝑆:On–1-1→On | ||
Theorem | fin1a2lem3 10397 | Lemma for fin1a2 10410. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
⊢ 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥)) ⇒ ⊢ (𝐴 ∈ ω → (𝐸‘𝐴) = (2o ·o 𝐴)) | ||
Theorem | fin1a2lem4 10398 | Lemma for fin1a2 10410. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
⊢ 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥)) ⇒ ⊢ 𝐸:ω–1-1→ω | ||
Theorem | fin1a2lem5 10399 | Lemma for fin1a2 10410. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
⊢ 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥)) ⇒ ⊢ (𝐴 ∈ ω → (𝐴 ∈ ran 𝐸 ↔ ¬ suc 𝐴 ∈ ran 𝐸)) | ||
Theorem | fin1a2lem6 10400 | Lemma for fin1a2 10410. 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 𝐸–1-1-onto→(ω ∖ ran 𝐸) |
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