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Theorem alexsubALTlem2 23876
Description: Lemma for alexsubALT 23879. Every subset of a base which has no finite subcover is a subset of a maximal such collection. (Contributed by Jeff Hankins, 27-Jan-2010.)
Hypothesis
Ref Expression
alexsubALT.1 𝑋 = 𝐽
Assertion
Ref Expression
alexsubALTlem2 (((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏) → ∃𝑢 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅})∀𝑣 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ¬ 𝑢𝑣)
Distinct variable groups:   𝑎,𝑏,𝑐,𝑑,𝑢,𝑣,𝑥,𝑧,𝐽   𝑋,𝑎,𝑏,𝑐,𝑑,𝑢,𝑣,𝑥,𝑧

Proof of Theorem alexsubALTlem2
Dummy variables 𝑛 𝑤 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3968 . . . . . . . . . . . . 13 (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) → (𝑤𝑦𝑤 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅})))
2 elun 4141 . . . . . . . . . . . . . . 15 (𝑤 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ↔ (𝑤 ∈ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∨ 𝑤 ∈ {∅}))
3 sseq2 4001 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑤 → (𝑎𝑧𝑎𝑤))
4 pweq 4609 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = 𝑤 → 𝒫 𝑧 = 𝒫 𝑤)
54ineq1d 4204 . . . . . . . . . . . . . . . . . . 19 (𝑧 = 𝑤 → (𝒫 𝑧 ∩ Fin) = (𝒫 𝑤 ∩ Fin))
65raleqdv 3317 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑤 → (∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏 ↔ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏))
73, 6anbi12d 630 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑤 → ((𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏) ↔ (𝑎𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏)))
87elrab 3676 . . . . . . . . . . . . . . . 16 (𝑤 ∈ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ↔ (𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏)))
9 velsn 4637 . . . . . . . . . . . . . . . 16 (𝑤 ∈ {∅} ↔ 𝑤 = ∅)
108, 9orbi12i 911 . . . . . . . . . . . . . . 15 ((𝑤 ∈ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∨ 𝑤 ∈ {∅}) ↔ ((𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏)) ∨ 𝑤 = ∅))
112, 10bitri 275 . . . . . . . . . . . . . 14 (𝑤 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ↔ ((𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏)) ∨ 𝑤 = ∅))
12 elpwi 4602 . . . . . . . . . . . . . . . 16 (𝑤 ∈ 𝒫 (fi‘𝑥) → 𝑤 ⊆ (fi‘𝑥))
1312adantr 480 . . . . . . . . . . . . . . 15 ((𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏)) → 𝑤 ⊆ (fi‘𝑥))
14 0ss 4389 . . . . . . . . . . . . . . . 16 ∅ ⊆ (fi‘𝑥)
15 sseq1 4000 . . . . . . . . . . . . . . . 16 (𝑤 = ∅ → (𝑤 ⊆ (fi‘𝑥) ↔ ∅ ⊆ (fi‘𝑥)))
1614, 15mpbiri 258 . . . . . . . . . . . . . . 15 (𝑤 = ∅ → 𝑤 ⊆ (fi‘𝑥))
1713, 16jaoi 854 . . . . . . . . . . . . . 14 (((𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏)) ∨ 𝑤 = ∅) → 𝑤 ⊆ (fi‘𝑥))
1811, 17sylbi 216 . . . . . . . . . . . . 13 (𝑤 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) → 𝑤 ⊆ (fi‘𝑥))
191, 18syl6 35 . . . . . . . . . . . 12 (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) → (𝑤𝑦𝑤 ⊆ (fi‘𝑥)))
2019ralrimiv 3137 . . . . . . . . . . 11 (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) → ∀𝑤𝑦 𝑤 ⊆ (fi‘𝑥))
21 unissb 4934 . . . . . . . . . . 11 ( 𝑦 ⊆ (fi‘𝑥) ↔ ∀𝑤𝑦 𝑤 ⊆ (fi‘𝑥))
2220, 21sylibr 233 . . . . . . . . . 10 (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) → 𝑦 ⊆ (fi‘𝑥))
2322adantr 480 . . . . . . . . 9 ((𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∧ [] Or 𝑦) → 𝑦 ⊆ (fi‘𝑥))
2423ad2antlr 724 . . . . . . . 8 (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∧ [] Or 𝑦)) ∧ ¬ 𝑦 = ∅) → 𝑦 ⊆ (fi‘𝑥))
25 vuniex 7723 . . . . . . . . 9 𝑦 ∈ V
2625elpw 4599 . . . . . . . 8 ( 𝑦 ∈ 𝒫 (fi‘𝑥) ↔ 𝑦 ⊆ (fi‘𝑥))
2724, 26sylibr 233 . . . . . . 7 (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∧ [] Or 𝑦)) ∧ ¬ 𝑦 = ∅) → 𝑦 ∈ 𝒫 (fi‘𝑥))
28 uni0b 4928 . . . . . . . . . 10 ( 𝑦 = ∅ ↔ 𝑦 ⊆ {∅})
2928notbii 320 . . . . . . . . 9 𝑦 = ∅ ↔ ¬ 𝑦 ⊆ {∅})
30 disjssun 4460 . . . . . . . . . . . . 13 ((𝑦 ∩ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)}) = ∅ → (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ↔ 𝑦 ⊆ {∅}))
3130biimpcd 248 . . . . . . . . . . . 12 (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) → ((𝑦 ∩ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)}) = ∅ → 𝑦 ⊆ {∅}))
3231necon3bd 2946 . . . . . . . . . . 11 (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) → (¬ 𝑦 ⊆ {∅} → (𝑦 ∩ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)}) ≠ ∅))
33 n0 4339 . . . . . . . . . . . 12 ((𝑦 ∩ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)}) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (𝑦 ∩ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)}))
34 elin 3957 . . . . . . . . . . . . . . 15 (𝑤 ∈ (𝑦 ∩ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)}) ↔ (𝑤𝑦𝑤 ∈ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)}))
358anbi2i 622 . . . . . . . . . . . . . . 15 ((𝑤𝑦𝑤 ∈ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)}) ↔ (𝑤𝑦 ∧ (𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏))))
3634, 35bitri 275 . . . . . . . . . . . . . 14 (𝑤 ∈ (𝑦 ∩ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)}) ↔ (𝑤𝑦 ∧ (𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏))))
37 simprrl 778 . . . . . . . . . . . . . . 15 ((𝑤𝑦 ∧ (𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏))) → 𝑎𝑤)
38 simpl 482 . . . . . . . . . . . . . . 15 ((𝑤𝑦 ∧ (𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏))) → 𝑤𝑦)
39 ssuni 4927 . . . . . . . . . . . . . . 15 ((𝑎𝑤𝑤𝑦) → 𝑎 𝑦)
4037, 38, 39syl2anc 583 . . . . . . . . . . . . . 14 ((𝑤𝑦 ∧ (𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏))) → 𝑎 𝑦)
4136, 40sylbi 216 . . . . . . . . . . . . 13 (𝑤 ∈ (𝑦 ∩ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)}) → 𝑎 𝑦)
4241exlimiv 1925 . . . . . . . . . . . 12 (∃𝑤 𝑤 ∈ (𝑦 ∩ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)}) → 𝑎 𝑦)
4333, 42sylbi 216 . . . . . . . . . . 11 ((𝑦 ∩ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)}) ≠ ∅ → 𝑎 𝑦)
4432, 43syl6 35 . . . . . . . . . 10 (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) → (¬ 𝑦 ⊆ {∅} → 𝑎 𝑦))
4544ad2antrl 725 . . . . . . . . 9 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∧ [] Or 𝑦)) → (¬ 𝑦 ⊆ {∅} → 𝑎 𝑦))
4629, 45biimtrid 241 . . . . . . . 8 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∧ [] Or 𝑦)) → (¬ 𝑦 = ∅ → 𝑎 𝑦))
4746imp 406 . . . . . . 7 (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∧ [] Or 𝑦)) ∧ ¬ 𝑦 = ∅) → 𝑎 𝑦)
48 elfpw 9351 . . . . . . . . . 10 (𝑛 ∈ (𝒫 𝑦 ∩ Fin) ↔ (𝑛 𝑦𝑛 ∈ Fin))
49 unieq 4911 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ∅ → 𝑦 = ∅)
50 uni0 4930 . . . . . . . . . . . . . . . . . . 19 ∅ = ∅
5149, 50eqtrdi 2780 . . . . . . . . . . . . . . . . . 18 (𝑦 = ∅ → 𝑦 = ∅)
5251necon3bi 2959 . . . . . . . . . . . . . . . . 17 𝑦 = ∅ → 𝑦 ≠ ∅)
5352adantr 480 . . . . . . . . . . . . . . . 16 ((¬ 𝑦 = ∅ ∧ 𝑛 ∈ Fin) → 𝑦 ≠ ∅)
5453ad2antrl 725 . . . . . . . . . . . . . . 15 (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∧ [] Or 𝑦)) ∧ ((¬ 𝑦 = ∅ ∧ 𝑛 ∈ Fin) ∧ 𝑛 𝑦)) → 𝑦 ≠ ∅)
55 simplrr 775 . . . . . . . . . . . . . . 15 (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∧ [] Or 𝑦)) ∧ ((¬ 𝑦 = ∅ ∧ 𝑛 ∈ Fin) ∧ 𝑛 𝑦)) → [] Or 𝑦)
56 simprlr 777 . . . . . . . . . . . . . . 15 (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∧ [] Or 𝑦)) ∧ ((¬ 𝑦 = ∅ ∧ 𝑛 ∈ Fin) ∧ 𝑛 𝑦)) → 𝑛 ∈ Fin)
57 simprr 770 . . . . . . . . . . . . . . 15 (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∧ [] Or 𝑦)) ∧ ((¬ 𝑦 = ∅ ∧ 𝑛 ∈ Fin) ∧ 𝑛 𝑦)) → 𝑛 𝑦)
58 finsschain 9356 . . . . . . . . . . . . . . 15 (((𝑦 ≠ ∅ ∧ [] Or 𝑦) ∧ (𝑛 ∈ Fin ∧ 𝑛 𝑦)) → ∃𝑤𝑦 𝑛𝑤)
5954, 55, 56, 57, 58syl22anc 836 . . . . . . . . . . . . . 14 (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∧ [] Or 𝑦)) ∧ ((¬ 𝑦 = ∅ ∧ 𝑛 ∈ Fin) ∧ 𝑛 𝑦)) → ∃𝑤𝑦 𝑛𝑤)
6059expr 456 . . . . . . . . . . . . 13 (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∧ [] Or 𝑦)) ∧ (¬ 𝑦 = ∅ ∧ 𝑛 ∈ Fin)) → (𝑛 𝑦 → ∃𝑤𝑦 𝑛𝑤))
61 0elpw 5345 . . . . . . . . . . . . . . . . . . . 20 ∅ ∈ 𝒫 𝑎
62 0fin 9168 . . . . . . . . . . . . . . . . . . . 20 ∅ ∈ Fin
6361, 62elini 4186 . . . . . . . . . . . . . . . . . . 19 ∅ ∈ (𝒫 𝑎 ∩ Fin)
64 unieq 4911 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = ∅ → 𝑏 = ∅)
6564eqeq2d 2735 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 = ∅ → (𝑋 = 𝑏𝑋 = ∅))
6665notbid 318 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = ∅ → (¬ 𝑋 = 𝑏 ↔ ¬ 𝑋 = ∅))
6766rspccv 3601 . . . . . . . . . . . . . . . . . . 19 (∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏 → (∅ ∈ (𝒫 𝑎 ∩ Fin) → ¬ 𝑋 = ∅))
6863, 67mpi 20 . . . . . . . . . . . . . . . . . 18 (∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏 → ¬ 𝑋 = ∅)
69 velpw 4600 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 ∈ 𝒫 𝑤𝑛𝑤)
70 elin 3957 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 ∈ (𝒫 𝑤 ∩ Fin) ↔ (𝑛 ∈ 𝒫 𝑤𝑛 ∈ Fin))
71 unieq 4911 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑏 = 𝑛 𝑏 = 𝑛)
7271eqeq2d 2735 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑏 = 𝑛 → (𝑋 = 𝑏𝑋 = 𝑛))
7372notbid 318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑏 = 𝑛 → (¬ 𝑋 = 𝑏 ↔ ¬ 𝑋 = 𝑛))
7473rspccv 3601 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏 → (𝑛 ∈ (𝒫 𝑤 ∩ Fin) → ¬ 𝑋 = 𝑛))
7570, 74biimtrrid 242 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏 → ((𝑛 ∈ 𝒫 𝑤𝑛 ∈ Fin) → ¬ 𝑋 = 𝑛))
7675expd 415 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏 → (𝑛 ∈ 𝒫 𝑤 → (𝑛 ∈ Fin → ¬ 𝑋 = 𝑛)))
7769, 76biimtrrid 242 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏 → (𝑛𝑤 → (𝑛 ∈ Fin → ¬ 𝑋 = 𝑛)))
7877com23 86 . . . . . . . . . . . . . . . . . . . . . . . 24 (∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏 → (𝑛 ∈ Fin → (𝑛𝑤 → ¬ 𝑋 = 𝑛)))
7978ad2antll 726 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏)) → (𝑛 ∈ Fin → (𝑛𝑤 → ¬ 𝑋 = 𝑛)))
8079a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 𝑋 = ∅ → ((𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏)) → (𝑛 ∈ Fin → (𝑛𝑤 → ¬ 𝑋 = 𝑛))))
81 sseq2 4001 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = ∅ → (𝑛𝑤𝑛 ⊆ ∅))
82 ss0 4391 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ⊆ ∅ → 𝑛 = ∅)
8381, 82syl6bi 253 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = ∅ → (𝑛𝑤𝑛 = ∅))
84 unieq 4911 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 = ∅ → 𝑛 = ∅)
8584eqeq2d 2735 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 = ∅ → (𝑋 = 𝑛𝑋 = ∅))
8685notbid 318 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = ∅ → (¬ 𝑋 = 𝑛 ↔ ¬ 𝑋 = ∅))
8786biimprcd 249 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑋 = ∅ → (𝑛 = ∅ → ¬ 𝑋 = 𝑛))
8887a1dd 50 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑋 = ∅ → (𝑛 = ∅ → (𝑛 ∈ Fin → ¬ 𝑋 = 𝑛)))
8983, 88syl9r 78 . . . . . . . . . . . . . . . . . . . . . . 23 𝑋 = ∅ → (𝑤 = ∅ → (𝑛𝑤 → (𝑛 ∈ Fin → ¬ 𝑋 = 𝑛))))
9089com34 91 . . . . . . . . . . . . . . . . . . . . . 22 𝑋 = ∅ → (𝑤 = ∅ → (𝑛 ∈ Fin → (𝑛𝑤 → ¬ 𝑋 = 𝑛))))
9180, 90jaod 856 . . . . . . . . . . . . . . . . . . . . 21 𝑋 = ∅ → (((𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏)) ∨ 𝑤 = ∅) → (𝑛 ∈ Fin → (𝑛𝑤 → ¬ 𝑋 = 𝑛))))
9211, 91biimtrid 241 . . . . . . . . . . . . . . . . . . . 20 𝑋 = ∅ → (𝑤 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) → (𝑛 ∈ Fin → (𝑛𝑤 → ¬ 𝑋 = 𝑛))))
931, 92sylan9r 508 . . . . . . . . . . . . . . . . . . 19 ((¬ 𝑋 = ∅ ∧ 𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅})) → (𝑤𝑦 → (𝑛 ∈ Fin → (𝑛𝑤 → ¬ 𝑋 = 𝑛))))
9493com23 86 . . . . . . . . . . . . . . . . . 18 ((¬ 𝑋 = ∅ ∧ 𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅})) → (𝑛 ∈ Fin → (𝑤𝑦 → (𝑛𝑤 → ¬ 𝑋 = 𝑛))))
9568, 94sylan 579 . . . . . . . . . . . . . . . . 17 ((∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅})) → (𝑛 ∈ Fin → (𝑤𝑦 → (𝑛𝑤 → ¬ 𝑋 = 𝑛))))
9695ad2ant2lr 745 . . . . . . . . . . . . . . . 16 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∧ [] Or 𝑦)) → (𝑛 ∈ Fin → (𝑤𝑦 → (𝑛𝑤 → ¬ 𝑋 = 𝑛))))
9796imp 406 . . . . . . . . . . . . . . 15 (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∧ [] Or 𝑦)) ∧ 𝑛 ∈ Fin) → (𝑤𝑦 → (𝑛𝑤 → ¬ 𝑋 = 𝑛)))
9897adantrl 713 . . . . . . . . . . . . . 14 (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∧ [] Or 𝑦)) ∧ (¬ 𝑦 = ∅ ∧ 𝑛 ∈ Fin)) → (𝑤𝑦 → (𝑛𝑤 → ¬ 𝑋 = 𝑛)))
9998rexlimdv 3145 . . . . . . . . . . . . 13 (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∧ [] Or 𝑦)) ∧ (¬ 𝑦 = ∅ ∧ 𝑛 ∈ Fin)) → (∃𝑤𝑦 𝑛𝑤 → ¬ 𝑋 = 𝑛))
10060, 99syld 47 . . . . . . . . . . . 12 (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∧ [] Or 𝑦)) ∧ (¬ 𝑦 = ∅ ∧ 𝑛 ∈ Fin)) → (𝑛 𝑦 → ¬ 𝑋 = 𝑛))
101100expr 456 . . . . . . . . . . 11 (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∧ [] Or 𝑦)) ∧ ¬ 𝑦 = ∅) → (𝑛 ∈ Fin → (𝑛 𝑦 → ¬ 𝑋 = 𝑛)))
102101impcomd 411 . . . . . . . . . 10 (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∧ [] Or 𝑦)) ∧ ¬ 𝑦 = ∅) → ((𝑛 𝑦𝑛 ∈ Fin) → ¬ 𝑋 = 𝑛))
10348, 102biimtrid 241 . . . . . . . . 9 (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∧ [] Or 𝑦)) ∧ ¬ 𝑦 = ∅) → (𝑛 ∈ (𝒫 𝑦 ∩ Fin) → ¬ 𝑋 = 𝑛))
104103ralrimiv 3137 . . . . . . . 8 (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∧ [] Or 𝑦)) ∧ ¬ 𝑦 = ∅) → ∀𝑛 ∈ (𝒫 𝑦 ∩ Fin) ¬ 𝑋 = 𝑛)
105 unieq 4911 . . . . . . . . . . 11 (𝑛 = 𝑏 𝑛 = 𝑏)
106105eqeq2d 2735 . . . . . . . . . 10 (𝑛 = 𝑏 → (𝑋 = 𝑛𝑋 = 𝑏))
107106notbid 318 . . . . . . . . 9 (𝑛 = 𝑏 → (¬ 𝑋 = 𝑛 ↔ ¬ 𝑋 = 𝑏))
108107cbvralvw 3226 . . . . . . . 8 (∀𝑛 ∈ (𝒫 𝑦 ∩ Fin) ¬ 𝑋 = 𝑛 ↔ ∀𝑏 ∈ (𝒫 𝑦 ∩ Fin) ¬ 𝑋 = 𝑏)
109104, 108sylib 217 . . . . . . 7 (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∧ [] Or 𝑦)) ∧ ¬ 𝑦 = ∅) → ∀𝑏 ∈ (𝒫 𝑦 ∩ Fin) ¬ 𝑋 = 𝑏)
11027, 47, 109jca32 515 . . . . . 6 (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∧ [] Or 𝑦)) ∧ ¬ 𝑦 = ∅) → ( 𝑦 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 𝑦 ∧ ∀𝑏 ∈ (𝒫 𝑦 ∩ Fin) ¬ 𝑋 = 𝑏)))
111110ex 412 . . . . 5 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∧ [] Or 𝑦)) → (¬ 𝑦 = ∅ → ( 𝑦 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 𝑦 ∧ ∀𝑏 ∈ (𝒫 𝑦 ∩ Fin) ¬ 𝑋 = 𝑏))))
112 orcom 867 . . . . . 6 (( 𝑦 ∈ {∅} ∨ 𝑦 ∈ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)}) ↔ ( 𝑦 ∈ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∨ 𝑦 ∈ {∅}))
11325elsn 4636 . . . . . . . 8 ( 𝑦 ∈ {∅} ↔ 𝑦 = ∅)
114 sseq2 4001 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝑎𝑧𝑎 𝑦))
115 pweq 4609 . . . . . . . . . . . 12 (𝑧 = 𝑦 → 𝒫 𝑧 = 𝒫 𝑦)
116115ineq1d 4204 . . . . . . . . . . 11 (𝑧 = 𝑦 → (𝒫 𝑧 ∩ Fin) = (𝒫 𝑦 ∩ Fin))
117116raleqdv 3317 . . . . . . . . . 10 (𝑧 = 𝑦 → (∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏 ↔ ∀𝑏 ∈ (𝒫 𝑦 ∩ Fin) ¬ 𝑋 = 𝑏))
118114, 117anbi12d 630 . . . . . . . . 9 (𝑧 = 𝑦 → ((𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏) ↔ (𝑎 𝑦 ∧ ∀𝑏 ∈ (𝒫 𝑦 ∩ Fin) ¬ 𝑋 = 𝑏)))
119118elrab 3676 . . . . . . . 8 ( 𝑦 ∈ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ↔ ( 𝑦 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 𝑦 ∧ ∀𝑏 ∈ (𝒫 𝑦 ∩ Fin) ¬ 𝑋 = 𝑏)))
120113, 119orbi12i 911 . . . . . . 7 (( 𝑦 ∈ {∅} ∨ 𝑦 ∈ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)}) ↔ ( 𝑦 = ∅ ∨ ( 𝑦 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 𝑦 ∧ ∀𝑏 ∈ (𝒫 𝑦 ∩ Fin) ¬ 𝑋 = 𝑏))))
121 df-or 845 . . . . . . 7 (( 𝑦 = ∅ ∨ ( 𝑦 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 𝑦 ∧ ∀𝑏 ∈ (𝒫 𝑦 ∩ Fin) ¬ 𝑋 = 𝑏))) ↔ (¬ 𝑦 = ∅ → ( 𝑦 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 𝑦 ∧ ∀𝑏 ∈ (𝒫 𝑦 ∩ Fin) ¬ 𝑋 = 𝑏))))
122120, 121bitr2i 276 . . . . . 6 ((¬ 𝑦 = ∅ → ( 𝑦 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 𝑦 ∧ ∀𝑏 ∈ (𝒫 𝑦 ∩ Fin) ¬ 𝑋 = 𝑏))) ↔ ( 𝑦 ∈ {∅} ∨ 𝑦 ∈ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)}))
123 elun 4141 . . . . . 6 ( 𝑦 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ↔ ( 𝑦 ∈ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∨ 𝑦 ∈ {∅}))
124112, 122, 1233bitr4i 303 . . . . 5 ((¬ 𝑦 = ∅ → ( 𝑦 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 𝑦 ∧ ∀𝑏 ∈ (𝒫 𝑦 ∩ Fin) ¬ 𝑋 = 𝑏))) ↔ 𝑦 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}))
125111, 124sylib 217 . . . 4 ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∧ [] Or 𝑦)) → 𝑦 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}))
126125ex 412 . . 3 (((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏) → ((𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∧ [] Or 𝑦) → 𝑦 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅})))
127126alrimiv 1922 . 2 (((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏) → ∀𝑦((𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∧ [] Or 𝑦) → 𝑦 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅})))
128 fvex 6895 . . . . . 6 (fi‘𝑥) ∈ V
129128pwex 5369 . . . . 5 𝒫 (fi‘𝑥) ∈ V
130129rabex 5323 . . . 4 {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∈ V
131 p0ex 5373 . . . 4 {∅} ∈ V
132130, 131unex 7727 . . 3 ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∈ V
133132zorn 10499 . 2 (∀𝑦((𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∧ [] Or 𝑦) → 𝑦 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅})) → ∃𝑢 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅})∀𝑣 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ¬ 𝑢𝑣)
134127, 133syl 17 1 (((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏) → ∃𝑢 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅})∀𝑣 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ¬ 𝑢𝑣)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 844  w3a 1084  wal 1531   = wceq 1533  wex 1773  wcel 2098  wne 2932  wral 3053  wrex 3062  {crab 3424  cun 3939  cin 3940  wss 3941  wpss 3942  c0 4315  𝒫 cpw 4595  {csn 4621   cuni 4900   Or wor 5578  cfv 6534   [] crpss 7706  Fincfn 8936  ficfi 9402  topGenctg 17384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-ac2 10455
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-se 5623  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-isom 6543  df-riota 7358  df-ov 7405  df-rpss 7707  df-om 7850  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-en 8937  df-fin 8940  df-card 9931  df-ac 10108
This theorem is referenced by:  alexsubALTlem4  23878
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