HomeHome Metamath Proof Explorer
Theorem List (p. 92 of 424)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-27745)
  Hilbert Space Explorer  Hilbert Space Explorer
(27746-29270)
  Users' Mathboxes  Users' Mathboxes
(29271-42316)
 

Theorem List for Metamath Proof Explorer - 9101-9200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremisfin2 9101* Definition of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
(𝐴𝑉 → (𝐴 ∈ FinII ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦)))
 
Theoremfin2i 9102 Property of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
(((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [] Or 𝐵)) → 𝐵𝐵)
 
Theoremisfin3 9103 Definition of a III-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
(𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV)
 
Theoremisfin4 9104* Definition of a IV-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
(𝐴𝑉 → (𝐴 ∈ FinIV ↔ ¬ ∃𝑦(𝑦𝐴𝑦𝐴)))
 
Theoremfin4i 9105 Infer that a set is IV-infinite. (Contributed by Stefan O'Rear, 16-May-2015.)
((𝑋𝐴𝑋𝐴) → ¬ 𝐴 ∈ FinIV)
 
Theoremisfin5 9106 Definition of a V-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
(𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 +𝑐 𝐴)))
 
Theoremisfin6 9107 Definition of a VI-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
(𝐴 ∈ FinVI ↔ (𝐴 ≺ 2𝑜𝐴 ≺ (𝐴 × 𝐴)))
 
Theoremisfin7 9108* Definition of a VII-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
(𝐴𝑉 → (𝐴 ∈ FinVII ↔ ¬ ∃𝑦 ∈ (On ∖ ω)𝐴𝑦))
 
Theoremsdom2en01 9109 A set with less than two elements has 0 or 1. (Contributed by Stefan O'Rear, 30-Oct-2014.)
(𝐴 ≺ 2𝑜 ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1𝑜))
 
Theoreminfpssrlem1 9110 Lemma for infpssr 9115. (Contributed by Stefan O'Rear, 30-Oct-2014.)
(𝜑𝐵𝐴)    &   (𝜑𝐹:𝐵1-1-onto𝐴)    &   (𝜑𝐶 ∈ (𝐴𝐵))    &   𝐺 = (rec(𝐹, 𝐶) ↾ ω)       (𝜑 → (𝐺‘∅) = 𝐶)
 
Theoreminfpssrlem2 9111 Lemma for infpssr 9115. (Contributed by Stefan O'Rear, 30-Oct-2014.)
(𝜑𝐵𝐴)    &   (𝜑𝐹:𝐵1-1-onto𝐴)    &   (𝜑𝐶 ∈ (𝐴𝐵))    &   𝐺 = (rec(𝐹, 𝐶) ↾ ω)       (𝑀 ∈ ω → (𝐺‘suc 𝑀) = (𝐹‘(𝐺𝑀)))
 
Theoreminfpssrlem3 9112 Lemma for infpssr 9115. (Contributed by Stefan O'Rear, 30-Oct-2014.)
(𝜑𝐵𝐴)    &   (𝜑𝐹:𝐵1-1-onto𝐴)    &   (𝜑𝐶 ∈ (𝐴𝐵))    &   𝐺 = (rec(𝐹, 𝐶) ↾ ω)       (𝜑𝐺:ω⟶𝐴)
 
Theoreminfpssrlem4 9113 Lemma for infpssr 9115. (Contributed by Stefan O'Rear, 30-Oct-2014.)
(𝜑𝐵𝐴)    &   (𝜑𝐹:𝐵1-1-onto𝐴)    &   (𝜑𝐶 ∈ (𝐴𝐵))    &   𝐺 = (rec(𝐹, 𝐶) ↾ ω)       ((𝜑𝑀 ∈ ω ∧ 𝑁𝑀) → (𝐺𝑀) ≠ (𝐺𝑁))
 
Theoreminfpssrlem5 9114 Lemma for infpssr 9115. (Contributed by Stefan O'Rear, 30-Oct-2014.)
(𝜑𝐵𝐴)    &   (𝜑𝐹:𝐵1-1-onto𝐴)    &   (𝜑𝐶 ∈ (𝐴𝐵))    &   𝐺 = (rec(𝐹, 𝐶) ↾ ω)       (𝜑 → (𝐴𝑉 → ω ≼ 𝐴))
 
Theoreminfpssr 9115 Dedekind infinity implies existence of a denumerable subset: take a single point witnessing the proper subset relation and iterate the embedding. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)
((𝑋𝐴𝑋𝐴) → ω ≼ 𝐴)
 
Theoremfin4en1 9116 Dedekind finite is a cardinal property. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)
(𝐴𝐵 → (𝐴 ∈ FinIV𝐵 ∈ FinIV))
 
Theoremssfin4 9117 Dedekind finite sets have Dedekind finite subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.) (Revised by Mario Carneiro, 6-May-2015.)
((𝐴 ∈ FinIV𝐵𝐴) → 𝐵 ∈ FinIV)
 
Theoremdomfin4 9118 A set dominated by a Dedekind finite set is Dedekind finite. (Contributed by Mario Carneiro, 16-May-2015.)
((𝐴 ∈ FinIV𝐵𝐴) → 𝐵 ∈ FinIV)
 
Theoremominf4 9119 ω is Dedekind infinite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Proof shortened by Mario Carneiro, 16-May-2015.)
¬ ω ∈ FinIV
 
TheoreminfpssALT 9120* Alternate proof of infpss 9024, shorter but requiring Replacement (ax-rep 4762). (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(ω ≼ 𝐴 → ∃𝑥(𝑥𝐴𝑥𝐴))
 
Theoremisfin4-2 9121 Alternate definition of IV-finite sets: they lack a denumerable subset. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)
(𝐴𝑉 → (𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝐴))
 
Theoremisfin4-3 9122 Alternate definition of IV-finite sets: they are strictly dominated by their successors. (Thus, the proper subset referred to in isfin4 9104 can be assumed to be only a singleton smaller than the original.) (Contributed by Mario Carneiro, 18-May-2015.)
(𝐴 ∈ FinIV𝐴 ≺ (𝐴 +𝑐 1𝑜))
 
Theoremfin23lem7 9123* Lemma for isfin2-2 9126. The componentwise complement of a nonempty collection of sets is nonempty. (Contributed by Stefan O'Rear, 31-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)
((𝐴𝑉𝐵 ⊆ 𝒫 𝐴𝐵 ≠ ∅) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝐴𝑥) ∈ 𝐵} ≠ ∅)
 
Theoremfin23lem11 9124* Lemma for isfin2-2 9126. (Contributed by Stefan O'Rear, 31-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)
(𝑧 = (𝐴𝑥) → (𝜓𝜒))    &   (𝑤 = (𝐴𝑣) → (𝜑𝜃))    &   ((𝑥𝐴𝑣𝐴) → (𝜒𝜃))       (𝐵 ⊆ 𝒫 𝐴 → (∃𝑥 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵}∀𝑤 ∈ {𝑐 ∈ 𝒫 𝐴 ∣ (𝐴𝑐) ∈ 𝐵} ¬ 𝜑 → ∃𝑧𝐵𝑣𝐵 ¬ 𝜓))
 
Theoremfin2i2 9125 A II-finite set contains minimal elements for every nonempty chain. (Contributed by Mario Carneiro, 16-May-2015.)
(((𝐴 ∈ FinII𝐵 ⊆ 𝒫 𝐴) ∧ (𝐵 ≠ ∅ ∧ [] Or 𝐵)) → 𝐵𝐵)
 
Theoremisfin2-2 9126* FinII expressed in terms of minimal elements. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Proof shortened by Mario Carneiro, 16-May-2015.)
(𝐴𝑉 → (𝐴 ∈ FinII ↔ ∀𝑦 ∈ 𝒫 𝒫 𝐴((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦)))
 
Theoremssfin2 9127 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 9128 II-finiteness is a cardinal property. (Contributed by Mario Carneiro, 18-May-2015.)
(𝐴𝐵 → (𝐴 ∈ FinII𝐵 ∈ FinII))
 
Theoremfin23lem24 9129 Lemma for fin23 9196. 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 9130 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 9131 Lemma for fin23 9196. 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 9132* Lemma for fin23lem22 9134. (Contributed by Stefan O'Rear, 1-Nov-2014.)
(((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃𝑗𝑆 (𝑗𝑆) ≈ 𝑖)
 
Theoremfin23lem23 9133* Lemma for fin23lem22 9134. (Contributed by Stefan O'Rear, 1-Nov-2014.)
(((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃!𝑗𝑆 (𝑗𝑆) ≈ 𝑖)
 
Theoremfin23lem22 9134* Lemma for fin23 9196 but could be used elsewhere if we find a good name for it. Explicit construction of a bijection (actually an isomorphism, see fin23lem27 9135) 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 9135* The mapping constructed in fin23lem22 9134 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 9136* 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 9137* 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 9138* 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 9139* Lemma for fin23 9196. 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 9140* Lemma for fin23 9196. 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 9141* Lemma for fin23 9196. 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 9142* Lemma for fin23 9196. 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 9143* Lemma for fin23 9196. 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 9144* Lemma for fin23 9196. 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 9145* Lemma for fin23 9196. 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 9146* Lemma for fin23 9196. 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 9147* Lemma for fin23 9196. 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 9148* Lemma for fin23 9196. 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 9149* Lemma for fin23 9196. 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 9150* Lemma for fin23 9196. 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 9151* Lemma for fin23 9196. 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 9152* Lemma for fin23 9196. 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 9153* Lemma for fin23 9196. 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 9154* Lemma for fin23 9196. 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 9155* Lemma for fin23 9196. 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 9156* Lemma for fin23 9196. 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 9157* Lemma for fin23 9196. 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 9158* Lemma for fin23 9196. 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 9159* Lemma for fin23 9196. 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 9160* Lemma for isfin3-2 9174. 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 9161* Lemma for isfin3-2 9174. 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 9162* Lemma for isfin3-2 9174. 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 9163* Lemma for isfin3-2 9174. 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 9164* Lemma for isfin3-2 9174. 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 9165* Lemma for isfin3-2 9174. 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 9166* Lemma for isfin3-2 9174. 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 9167* Lemma for isfin3-2 9174. 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 9168* Lemma for isfin3-2 9174. 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 9169* 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 9170* Lemma for isfin3-2 9174. Remove hypotheses from isf32lem10 9169. (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 9171* Lemma for isfin3-2 9174. (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 9172 One half of isfin3-2 9174. (Contributed by Mario Carneiro, 3-Jun-2015.)
(𝐴 ∈ FinIII → ¬ ω ≼* 𝐴)
 
Theoremisf33lem 9173* Lemma for isfin3-3 9175. (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 9174 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 9175* 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 9176* Inference from isfin3-3 9175. (This is actually a bit stronger than isfin3-3 9175 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 9177* Lemma for compsscnv 9178. (Contributed by Mario Carneiro, 17-May-2015.)
((𝑥 ∈ 𝒫 𝐴𝑦 = (𝐴𝑥)) → (𝑦 ∈ 𝒫 𝐴𝑥 = (𝐴𝑦)))
 
Theoremcompsscnv 9178* Complementation on a power set lattice is an involution. (Contributed by Mario Carneiro, 17-May-2015.)
𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))       𝐹 = 𝐹
 
Theoremisf34lem1 9179* Lemma for isfin3-4 9189. (Contributed by Stefan O'Rear, 7-Nov-2014.)
𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))       ((𝐴𝑉𝑋𝐴) → (𝐹𝑋) = (𝐴𝑋))
 
Theoremisf34lem2 9180* Lemma for isfin3-4 9189. (Contributed by Stefan O'Rear, 7-Nov-2014.)
𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))       (𝐴𝑉𝐹:𝒫 𝐴⟶𝒫 𝐴)
 
Theoremcompssiso 9181* 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 9182* Lemma for isfin3-4 9189. (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 9183* 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 9184* Lemma for isfin3-4 9189. (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 9185* Lemma for isfin3-4 9189. (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 9186* Lemma for isfin3-4 9189. (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 9187* Lemma for isfin3-4 9189. (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 9188* Inference from isfin3-4 9189. (Contributed by Mario Carneiro, 17-May-2015.)
((𝐴 ∈ FinIII𝐺:ω⟶𝒫 𝐴 ∧ ∀𝑥 ∈ ω (𝐺𝑥) ⊆ (𝐺‘suc 𝑥)) → ran 𝐺 ∈ ran 𝐺)
 
Theoremisfin3-4 9189* 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 9190 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 9191 Ia-finiteness is a cardinal property. (Contributed by Mario Carneiro, 18-May-2015.)
(𝐴𝐵 → (𝐴 ∈ FinIa𝐵 ∈ FinIa))
 
Theoremisfin1-2 9192 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 9193 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 9194 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 9195 Compact quantifier-free version of the standard definition df-fin 7944. (Contributed by Stefan O'Rear, 6-Jan-2015.)
Fin = ( ≈ “ ω)
 
Theoremfin23 9196 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 9125) 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 9253 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 9197 Every III-finite set is IV-finite. (Contributed by Stefan O'Rear, 30-Oct-2014.)
(𝐴 ∈ FinIII𝐴 ∈ FinIV)
 
Theoremisfin5-2 9198 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 9199 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 9200 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)
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42316
  Copyright terms: Public domain < Previous  Next >