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Theorem alexsubALTlem2 22344
Description: Lemma for alexsubALT 22347. 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 3889 . . . . . . . . . . . . 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 4052 . . . . . . . . . . . . . . 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 3920 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑤 → (𝑎𝑧𝑎𝑤))
4 pweq 4462 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = 𝑤 → 𝒫 𝑧 = 𝒫 𝑤)
54ineq1d 4114 . . . . . . . . . . . . . . . . . . 19 (𝑧 = 𝑤 → (𝒫 𝑧 ∩ Fin) = (𝒫 𝑤 ∩ Fin))
65raleqdv 3377 . . . . . . . . . . . . . . . . . 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 3621 . . . . . . . . . . . . . . . 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 4494 . . . . . . . . . . . . . . . 16 (𝑤 ∈ {∅} ↔ 𝑤 = ∅)
108, 9orbi12i 909 . . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . 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 4469 . . . . . . . . . . . . . . . 16 (𝑤 ∈ 𝒫 (fi‘𝑥) → 𝑤 ⊆ (fi‘𝑥))
1312adantr 481 . . . . . . . . . . . . . . 15 ((𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏)) → 𝑤 ⊆ (fi‘𝑥))
14 0ss 4276 . . . . . . . . . . . . . . . 16 ∅ ⊆ (fi‘𝑥)
15 sseq1 3919 . . . . . . . . . . . . . . . 16 (𝑤 = ∅ → (𝑤 ⊆ (fi‘𝑥) ↔ ∅ ⊆ (fi‘𝑥)))
1614, 15mpbiri 259 . . . . . . . . . . . . . . 15 (𝑤 = ∅ → 𝑤 ⊆ (fi‘𝑥))
1713, 16jaoi 852 . . . . . . . . . . . . . 14 (((𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏)) ∨ 𝑤 = ∅) → 𝑤 ⊆ (fi‘𝑥))
1811, 17sylbi 218 . . . . . . . . . . . . 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 3150 . . . . . . . . . . 11 (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) → ∀𝑤𝑦 𝑤 ⊆ (fi‘𝑥))
21 unissb 4782 . . . . . . . . . . 11 ( 𝑦 ⊆ (fi‘𝑥) ↔ ∀𝑤𝑦 𝑤 ⊆ (fi‘𝑥))
2220, 21sylibr 235 . . . . . . . . . 10 (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) → 𝑦 ⊆ (fi‘𝑥))
2322adantr 481 . . . . . . . . 9 ((𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∧ [] Or 𝑦) → 𝑦 ⊆ (fi‘𝑥))
2423ad2antlr 723 . . . . . . . 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 7331 . . . . . . . . 9 𝑦 ∈ V
2625elpw 4465 . . . . . . . 8 ( 𝑦 ∈ 𝒫 (fi‘𝑥) ↔ 𝑦 ⊆ (fi‘𝑥))
2724, 26sylibr 235 . . . . . . 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 4776 . . . . . . . . . 10 ( 𝑦 = ∅ ↔ 𝑦 ⊆ {∅})
2928notbii 321 . . . . . . . . 9 𝑦 = ∅ ↔ ¬ 𝑦 ⊆ {∅})
30 disjssun 4337 . . . . . . . . . . . . 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 250 . . . . . . . . . . . 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 3000 . . . . . . . . . . 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 4236 . . . . . . . . . . . 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 4096 . . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . 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 777 . . . . . . . . . . . . . . 15 ((𝑤𝑦 ∧ (𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏))) → 𝑎𝑤)
38 simpl 483 . . . . . . . . . . . . . . 15 ((𝑤𝑦 ∧ (𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏))) → 𝑤𝑦)
39 ssuni 4773 . . . . . . . . . . . . . . 15 ((𝑎𝑤𝑤𝑦) → 𝑎 𝑦)
4037, 38, 39syl2anc 584 . . . . . . . . . . . . . 14 ((𝑤𝑦 ∧ (𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏))) → 𝑎 𝑦)
4136, 40sylbi 218 . . . . . . . . . . . . 13 (𝑤 ∈ (𝑦 ∩ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)}) → 𝑎 𝑦)
4241exlimiv 1912 . . . . . . . . . . . 12 (∃𝑤 𝑤 ∈ (𝑦 ∩ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)}) → 𝑎 𝑦)
4333, 42sylbi 218 . . . . . . . . . . 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 724 . . . . . . . . 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, 45syl5bi 243 . . . . . . . 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 407 . . . . . . 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 8679 . . . . . . . . . 10 (𝑛 ∈ (𝒫 𝑦 ∩ Fin) ↔ (𝑛 𝑦𝑛 ∈ Fin))
49 unieq 4759 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ∅ → 𝑦 = ∅)
50 uni0 4778 . . . . . . . . . . . . . . . . . . 19 ∅ = ∅
5149, 50syl6eq 2849 . . . . . . . . . . . . . . . . . 18 (𝑦 = ∅ → 𝑦 = ∅)
5251necon3bi 3012 . . . . . . . . . . . . . . . . 17 𝑦 = ∅ → 𝑦 ≠ ∅)
5352adantr 481 . . . . . . . . . . . . . . . 16 ((¬ 𝑦 = ∅ ∧ 𝑛 ∈ Fin) → 𝑦 ≠ ∅)
5453ad2antrl 724 . . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . 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 8684 . . . . . . . . . . . . . . 15 (((𝑦 ≠ ∅ ∧ [] Or 𝑦) ∧ (𝑛 ∈ Fin ∧ 𝑛 𝑦)) → ∃𝑤𝑦 𝑛𝑤)
5954, 55, 56, 57, 58syl22anc 835 . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . 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 5154 . . . . . . . . . . . . . . . . . . . 20 ∅ ∈ 𝒫 𝑎
62 0fin 8599 . . . . . . . . . . . . . . . . . . . 20 ∅ ∈ Fin
6361, 62elini 4097 . . . . . . . . . . . . . . . . . . 19 ∅ ∈ (𝒫 𝑎 ∩ Fin)
64 unieq 4759 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = ∅ → 𝑏 = ∅)
6564eqeq2d 2807 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 = ∅ → (𝑋 = 𝑏𝑋 = ∅))
6665notbid 319 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = ∅ → (¬ 𝑋 = 𝑏 ↔ ¬ 𝑋 = ∅))
6766rspccv 3558 . . . . . . . . . . . . . . . . . . 19 (∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏 → (∅ ∈ (𝒫 𝑎 ∩ Fin) → ¬ 𝑋 = ∅))
6863, 67mpi 20 . . . . . . . . . . . . . . . . . 18 (∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏 → ¬ 𝑋 = ∅)
69 selpw 4466 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 ∈ 𝒫 𝑤𝑛𝑤)
70 elin 4096 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 ∈ (𝒫 𝑤 ∩ Fin) ↔ (𝑛 ∈ 𝒫 𝑤𝑛 ∈ Fin))
71 unieq 4759 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑏 = 𝑛 𝑏 = 𝑛)
7271eqeq2d 2807 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑏 = 𝑛 → (𝑋 = 𝑏𝑋 = 𝑛))
7372notbid 319 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑏 = 𝑛 → (¬ 𝑋 = 𝑏 ↔ ¬ 𝑋 = 𝑛))
7473rspccv 3558 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏 → (𝑛 ∈ (𝒫 𝑤 ∩ Fin) → ¬ 𝑋 = 𝑛))
7570, 74syl5bir 244 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏 → ((𝑛 ∈ 𝒫 𝑤𝑛 ∈ Fin) → ¬ 𝑋 = 𝑛))
7675expd 416 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏 → (𝑛 ∈ 𝒫 𝑤 → (𝑛 ∈ Fin → ¬ 𝑋 = 𝑛)))
7769, 76syl5bir 244 . . . . . . . . . . . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . . . . . . . . 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 3920 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = ∅ → (𝑛𝑤𝑛 ⊆ ∅))
82 ss0 4278 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ⊆ ∅ → 𝑛 = ∅)
8381, 82syl6bi 254 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = ∅ → (𝑛𝑤𝑛 = ∅))
84 unieq 4759 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 = ∅ → 𝑛 = ∅)
8584eqeq2d 2807 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 = ∅ → (𝑋 = 𝑛𝑋 = ∅))
8685notbid 319 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = ∅ → (¬ 𝑋 = 𝑛 ↔ ¬ 𝑋 = ∅))
8786biimprcd 251 . . . . . . . . . . . . . . . . . . . . . . . . 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 854 . . . . . . . . . . . . . . . . . . . . 21 𝑋 = ∅ → (((𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏)) ∨ 𝑤 = ∅) → (𝑛 ∈ Fin → (𝑛𝑤 → ¬ 𝑋 = 𝑛))))
9211, 91syl5bi 243 . . . . . . . . . . . . . . . . . . . 20 𝑋 = ∅ → (𝑤 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) → (𝑛 ∈ Fin → (𝑛𝑤 → ¬ 𝑋 = 𝑛))))
931, 92sylan9r 509 . . . . . . . . . . . . . . . . . . 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 580 . . . . . . . . . . . . . . . . 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 744 . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . 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 712 . . . . . . . . . . . . . 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 3248 . . . . . . . . . . . . 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 457 . . . . . . . . . . 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 412 . . . . . . . . . 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, 102syl5bi 243 . . . . . . . . 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 3150 . . . . . . . 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 4759 . . . . . . . . . . 11 (𝑛 = 𝑏 𝑛 = 𝑏)
106105eqeq2d 2807 . . . . . . . . . 10 (𝑛 = 𝑏 → (𝑋 = 𝑛𝑋 = 𝑏))
107106notbid 319 . . . . . . . . 9 (𝑛 = 𝑏 → (¬ 𝑋 = 𝑛 ↔ ¬ 𝑋 = 𝑏))
108107cbvralv 3405 . . . . . . . 8 (∀𝑛 ∈ (𝒫 𝑦 ∩ Fin) ¬ 𝑋 = 𝑛 ↔ ∀𝑏 ∈ (𝒫 𝑦 ∩ Fin) ¬ 𝑋 = 𝑏)
109104, 108sylib 219 . . . . . . 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 516 . . . . . 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 413 . . . . 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 865 . . . . . 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 4493 . . . . . . . 8 ( 𝑦 ∈ {∅} ↔ 𝑦 = ∅)
114 sseq2 3920 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝑎𝑧𝑎 𝑦))
115 pweq 4462 . . . . . . . . . . . 12 (𝑧 = 𝑦 → 𝒫 𝑧 = 𝒫 𝑦)
116115ineq1d 4114 . . . . . . . . . . 11 (𝑧 = 𝑦 → (𝒫 𝑧 ∩ Fin) = (𝒫 𝑦 ∩ Fin))
117116raleqdv 3377 . . . . . . . . . 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 3621 . . . . . . . 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 909 . . . . . . 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 843 . . . . . . 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 277 . . . . . 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 4052 . . . . . 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 304 . . . . 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 219 . . . 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 413 . . 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 1909 . 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 6558 . . . . . 6 (fi‘𝑥) ∈ V
129128pwex 5179 . . . . 5 𝒫 (fi‘𝑥) ∈ V
130129rabex 5133 . . . 4 {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∈ V
131 p0ex 5182 . . . 4 {∅} ∈ V
132130, 131unex 7333 . . 3 ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∈ V
133132zorn 9782 . 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 396  wo 842  w3a 1080  wal 1523   = wceq 1525  wex 1765  wcel 2083  wne 2986  wral 3107  wrex 3108  {crab 3111  cun 3863  cin 3864  wss 3865  wpss 3866  c0 4217  𝒫 cpw 4459  {csn 4478   cuni 4751   Or wor 5368  cfv 6232   [] crpss 7313  Fincfn 8364  ficfi 8727  topGenctg 16544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-ac2 9738
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-se 5410  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-isom 6241  df-riota 6984  df-rpss 7314  df-om 7444  df-wrecs 7805  df-recs 7867  df-1o 7960  df-er 8146  df-en 8365  df-fin 8368  df-card 9221  df-ac 9395
This theorem is referenced by:  alexsubALTlem4  22346
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