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Type | Label | Description |
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Statement | ||
Definition | df-fin2 10301* | A set is II-finite (Tarski finite) iff every nonempty chain of subsets contains a maximum element. Definition II of [Levy58] p. 2. (Contributed by Stefan O'Rear, 12-Nov-2014.) |
⊢ FinII = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝒫 𝑥((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦)} | ||
Definition | df-fin4 10302* | A set is IV-finite (Dedekind finite) iff it has no equinumerous proper subset. Definition IV of [Levy58] p. 3. (Contributed by Stefan O'Rear, 12-Nov-2014.) |
⊢ FinIV = {𝑥 ∣ ¬ ∃𝑦(𝑦 ⊊ 𝑥 ∧ 𝑦 ≈ 𝑥)} | ||
Definition | df-fin3 10303 | A set is III-finite (weakly Dedekind finite) iff its power set is Dedekind finite. Definition III of [Levy58] p. 2. (Contributed by Stefan O'Rear, 12-Nov-2014.) |
⊢ FinIII = {𝑥 ∣ 𝒫 𝑥 ∈ FinIV} | ||
Definition | df-fin5 10304 | A set is V-finite iff it behaves finitely under ⊔. Definition V of [Levy58] p. 3. (Contributed by Stefan O'Rear, 12-Nov-2014.) |
⊢ FinV = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥))} | ||
Definition | df-fin6 10305 | 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 ≺ 2o ∨ 𝑥 ≺ (𝑥 × 𝑥))} | ||
Definition | df-fin7 10306* | 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)𝑥 ≈ 𝑦} | ||
Theorem | isfin1a 10307* | Definition of a Ia-finite set. (Contributed by Stefan O'Rear, 16-May-2015.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIa ↔ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ Fin ∨ (𝐴 ∖ 𝑦) ∈ Fin))) | ||
Theorem | fin1ai 10308 | Property of a Ia-finite set. (Contributed by Stefan O'Rear, 16-May-2015.) |
⊢ ((𝐴 ∈ FinIa ∧ 𝑋 ⊆ 𝐴) → (𝑋 ∈ Fin ∨ (𝐴 ∖ 𝑋) ∈ Fin)) | ||
Theorem | isfin2 10309* | Definition of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinII ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ 𝑦))) | ||
Theorem | fin2i 10310 | Property of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.) |
⊢ (((𝐴 ∈ FinII ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [⊊] Or 𝐵)) → ∪ 𝐵 ∈ 𝐵) | ||
Theorem | isfin3 10311 | Definition of a III-finite set. (Contributed by Stefan O'Rear, 16-May-2015.) |
⊢ (𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV) | ||
Theorem | isfin4 10312* | Definition of a IV-finite set. (Contributed by Stefan O'Rear, 16-May-2015.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIV ↔ ¬ ∃𝑦(𝑦 ⊊ 𝐴 ∧ 𝑦 ≈ 𝐴))) | ||
Theorem | fin4i 10313 | Infer that a set is IV-infinite. (Contributed by Stefan O'Rear, 16-May-2015.) |
⊢ ((𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴) → ¬ 𝐴 ∈ FinIV) | ||
Theorem | isfin5 10314 | Definition of a V-finite set. (Contributed by Stefan O'Rear, 16-May-2015.) |
⊢ (𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴))) | ||
Theorem | isfin6 10315 | Definition of a VI-finite set. (Contributed by Stefan O'Rear, 16-May-2015.) |
⊢ (𝐴 ∈ FinVI ↔ (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴))) | ||
Theorem | isfin7 10316* | Definition of a VII-finite set. (Contributed by Stefan O'Rear, 16-May-2015.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinVII ↔ ¬ ∃𝑦 ∈ (On ∖ ω)𝐴 ≈ 𝑦)) | ||
Theorem | sdom2en01 10317 | A set with less than two elements has 0 or 1. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
⊢ (𝐴 ≺ 2o ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1o)) | ||
Theorem | infpssrlem1 10318 | Lemma for infpssr 10323. (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 → (𝐺‘∅) = 𝐶) | ||
Theorem | infpssrlem2 10319 | Lemma for infpssr 10323. (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) ⇒ ⊢ (𝑀 ∈ ω → (𝐺‘suc 𝑀) = (◡𝐹‘(𝐺‘𝑀))) | ||
Theorem | infpssrlem3 10320 | Lemma for infpssr 10323. (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 → 𝐺:ω⟶𝐴) | ||
Theorem | infpssrlem4 10321 | Lemma for infpssr 10323. (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 10322 | Lemma for infpssr 10323. (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 10323 | 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 10324 | 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 10325 | 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 10326 | 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 10327 | ω is Dedekind infinite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Proof shortened by Mario Carneiro, 16-May-2015.) |
⊢ ¬ ω ∈ FinIV | ||
Theorem | infpssALT 10328* | Alternate proof of infpss 10232, shorter but requiring Replacement (ax-rep 5279). (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 10329 | 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 10330 | Alternate definition of IV-finite sets: they are strictly dominated by their successors. (Thus, the proper subset referred to in isfin4 10312 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 10331* | Lemma for isfin2-2 10334. 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 10332* | Lemma for isfin2-2 10334. (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 10333 | 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 10334* | 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 10335 | 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 10336 | II-finiteness is a cardinal property. (Contributed by Mario Carneiro, 18-May-2015.) |
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ FinII → 𝐵 ∈ FinII)) | ||
Theorem | fin23lem24 10337 | Lemma for fin23 10404. 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 10338 | 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 10339 | Lemma for fin23 10404. 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 10340* | Lemma for fin23lem22 10342. (Contributed by Stefan O'Rear, 1-Nov-2014.) |
⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖) | ||
Theorem | fin23lem23 10341* | Lemma for fin23lem22 10342. (Contributed by Stefan O'Rear, 1-Nov-2014.) |
⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃!𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖) | ||
Theorem | fin23lem22 10342* | Lemma for fin23 10404 but could be used elsewhere if we find a good name for it. Explicit construction of a bijection (actually an isomorphism, see fin23lem27 10343) 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 10343* | The mapping constructed in fin23lem22 10342 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 10344* | 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 10345* | 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 10346* |
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 10347* | Lemma for fin23 10404. 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 10348* | Lemma for fin23 10404. 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 10349* | Lemma for fin23 10404. 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 10350* | Lemma for fin23 10404. 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 10351* | Lemma for fin23 10404. 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 10352* | Lemma for fin23 10404. 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 10353* | Lemma for fin23 10404. 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 10354* | Lemma for fin23 10404. 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 10355* | Lemma for fin23 10404. 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 10356* | Lemma for fin23 10404. 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 10357* | Lemma for fin23 10404. 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 10358* | Lemma for fin23 10404. 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 10359* | Lemma for fin23 10404. 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 10360* | Lemma for fin23 10404. 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 10361* | Lemma for fin23 10404. 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 10362* | Lemma for fin23 10404. 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 10363* | Lemma for fin23 10404. 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 10364* | Lemma for fin23 10404. 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 10365* | Lemma for fin23 10404. 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 10366* | Lemma for fin23 10404. 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 10367* | Lemma for fin23 10404. 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 10368* | Lemma for isfin3-2 10382. 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 10369* | Lemma for isfin3-2 10382. 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 10370* | Lemma for isfin3-2 10382. 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 10371* | Lemma for isfin3-2 10382. 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 10372* | Lemma for isfin3-2 10382. 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 10373* | Lemma for isfin3-2 10382. 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 10374* | Lemma for isfin3-2 10382. 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 10375* | Lemma for isfin3-2 10382. 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 10376* | Lemma for isfin3-2 10382. 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 10377* | 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 10378* | Lemma for isfin3-2 10382. Remove hypotheses from isf32lem10 10377. (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 10379* | Lemma for isfin3-2 10382. (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 10380 | One half of isfin3-2 10382. (Contributed by Mario Carneiro, 3-Jun-2015.) |
⊢ (𝐴 ∈ FinIII → ¬ ω ≼* 𝐴) | ||
Theorem | isf33lem 10381* | Lemma for isfin3-3 10383. (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 10382 | 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 10383* | 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 10384* | Inference from isfin3-3 10383. (This is actually a bit stronger than isfin3-3 10383 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 10385* | Lemma for compsscnv 10386. (Contributed by Mario Carneiro, 17-May-2015.) |
⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))) | ||
Theorem | compsscnv 10386* | 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 10387* | Lemma for isfin3-4 10397. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝐹‘𝑋) = (𝐴 ∖ 𝑋)) | ||
Theorem | isf34lem2 10388* | Lemma for isfin3-4 10397. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐹:𝒫 𝐴⟶𝒫 𝐴) | ||
Theorem | compssiso 10389* | 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 10390* | Lemma for isfin3-4 10397. (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 10391* | 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 10392* | Lemma for isfin3-4 10397. (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 10393* | Lemma for isfin3-4 10397. (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 10394* | Lemma for isfin3-4 10397. (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 10395* | Lemma for isfin3-4 10397. (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 10396* | Inference from isfin3-4 10397. (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 10397* | 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 10398 | 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 10399 | Ia-finiteness is a cardinal property. (Contributed by Mario Carneiro, 18-May-2015.) |
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ FinIa → 𝐵 ∈ FinIa)) | ||
Theorem | isfin1-2 10400 | 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) |
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