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Theorem alexsubALTlem2 22134
Description: Lemma for alexsubALT 22137. 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 3757 . . . . . . . . . . . . 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 3917 . . . . . . . . . . . . . . 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 3789 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑤 → (𝑎𝑧𝑎𝑤))
4 pweq 4320 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = 𝑤 → 𝒫 𝑧 = 𝒫 𝑤)
54ineq1d 3977 . . . . . . . . . . . . . . . . . . 19 (𝑧 = 𝑤 → (𝒫 𝑧 ∩ Fin) = (𝒫 𝑤 ∩ Fin))
65raleqdv 3292 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑤 → (∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏 ↔ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏))
73, 6anbi12d 624 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑤 → ((𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏) ↔ (𝑎𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏)))
87elrab 3521 . . . . . . . . . . . . . . . 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 4352 . . . . . . . . . . . . . . . 16 (𝑤 ∈ {∅} ↔ 𝑤 = ∅)
108, 9orbi12i 938 . . . . . . . . . . . . . . 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 266 . . . . . . . . . . . . . 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 4327 . . . . . . . . . . . . . . . 16 (𝑤 ∈ 𝒫 (fi‘𝑥) → 𝑤 ⊆ (fi‘𝑥))
1312adantr 472 . . . . . . . . . . . . . . 15 ((𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏)) → 𝑤 ⊆ (fi‘𝑥))
14 0ss 4136 . . . . . . . . . . . . . . . 16 ∅ ⊆ (fi‘𝑥)
15 sseq1 3788 . . . . . . . . . . . . . . . 16 (𝑤 = ∅ → (𝑤 ⊆ (fi‘𝑥) ↔ ∅ ⊆ (fi‘𝑥)))
1614, 15mpbiri 249 . . . . . . . . . . . . . . 15 (𝑤 = ∅ → 𝑤 ⊆ (fi‘𝑥))
1713, 16jaoi 883 . . . . . . . . . . . . . 14 (((𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏)) ∨ 𝑤 = ∅) → 𝑤 ⊆ (fi‘𝑥))
1811, 17sylbi 208 . . . . . . . . . . . . 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 3112 . . . . . . . . . . 11 (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) → ∀𝑤𝑦 𝑤 ⊆ (fi‘𝑥))
21 unissb 4629 . . . . . . . . . . 11 ( 𝑦 ⊆ (fi‘𝑥) ↔ ∀𝑤𝑦 𝑤 ⊆ (fi‘𝑥))
2220, 21sylibr 225 . . . . . . . . . 10 (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) → 𝑦 ⊆ (fi‘𝑥))
2322adantr 472 . . . . . . . . 9 ((𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∧ [] Or 𝑦) → 𝑦 ⊆ (fi‘𝑥))
2423ad2antlr 718 . . . . . . . 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 7154 . . . . . . . . 9 𝑦 ∈ V
2625elpw 4323 . . . . . . . 8 ( 𝑦 ∈ 𝒫 (fi‘𝑥) ↔ 𝑦 ⊆ (fi‘𝑥))
2724, 26sylibr 225 . . . . . . 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 4623 . . . . . . . . . 10 ( 𝑦 = ∅ ↔ 𝑦 ⊆ {∅})
2928notbii 311 . . . . . . . . 9 𝑦 = ∅ ↔ ¬ 𝑦 ⊆ {∅})
30 disjssun 4198 . . . . . . . . . . . . 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 240 . . . . . . . . . . . 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 2951 . . . . . . . . . . 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 4097 . . . . . . . . . . . 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 3960 . . . . . . . . . . . . . . 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 616 . . . . . . . . . . . . . . 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 266 . . . . . . . . . . . . . 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 799 . . . . . . . . . . . . . . 15 ((𝑤𝑦 ∧ (𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏))) → 𝑎𝑤)
38 simpl 474 . . . . . . . . . . . . . . 15 ((𝑤𝑦 ∧ (𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏))) → 𝑤𝑦)
39 ssuni 4620 . . . . . . . . . . . . . . 15 ((𝑎𝑤𝑤𝑦) → 𝑎 𝑦)
4037, 38, 39syl2anc 579 . . . . . . . . . . . . . 14 ((𝑤𝑦 ∧ (𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏))) → 𝑎 𝑦)
4136, 40sylbi 208 . . . . . . . . . . . . 13 (𝑤 ∈ (𝑦 ∩ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)}) → 𝑎 𝑦)
4241exlimiv 2025 . . . . . . . . . . . 12 (∃𝑤 𝑤 ∈ (𝑦 ∩ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)}) → 𝑎 𝑦)
4333, 42sylbi 208 . . . . . . . . . . 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 719 . . . . . . . . 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 233 . . . . . . . 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 395 . . . . . . 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 8477 . . . . . . . . . 10 (𝑛 ∈ (𝒫 𝑦 ∩ Fin) ↔ (𝑛 𝑦𝑛 ∈ Fin))
49 unieq 4604 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = ∅ → 𝑦 = ∅)
50 uni0 4625 . . . . . . . . . . . . . . . . . . . 20 ∅ = ∅
5149, 50syl6eq 2815 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ∅ → 𝑦 = ∅)
5251necon3bi 2963 . . . . . . . . . . . . . . . . . 18 𝑦 = ∅ → 𝑦 ≠ ∅)
5352adantr 472 . . . . . . . . . . . . . . . . 17 ((¬ 𝑦 = ∅ ∧ 𝑛 ∈ Fin) → 𝑦 ≠ ∅)
5453ad2antrl 719 . . . . . . . . . . . . . . . 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)) ∧ ((¬ 𝑦 = ∅ ∧ 𝑛 ∈ Fin) ∧ 𝑛 𝑦)) → 𝑦 ≠ ∅)
55 simplrr 796 . . . . . . . . . . . . . . . 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)) ∧ ((¬ 𝑦 = ∅ ∧ 𝑛 ∈ Fin) ∧ 𝑛 𝑦)) → [] Or 𝑦)
56 simprlr 798 . . . . . . . . . . . . . . . 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)) ∧ ((¬ 𝑦 = ∅ ∧ 𝑛 ∈ Fin) ∧ 𝑛 𝑦)) → 𝑛 ∈ Fin)
57 simprr 789 . . . . . . . . . . . . . . . 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)) ∧ ((¬ 𝑦 = ∅ ∧ 𝑛 ∈ Fin) ∧ 𝑛 𝑦)) → 𝑛 𝑦)
58 finsschain 8482 . . . . . . . . . . . . . . . 16 (((𝑦 ≠ ∅ ∧ [] Or 𝑦) ∧ (𝑛 ∈ Fin ∧ 𝑛 𝑦)) → ∃𝑤𝑦 𝑛𝑤)
5954, 55, 56, 57, 58syl22anc 867 . . . . . . . . . . . . . . 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)) → ∃𝑤𝑦 𝑛𝑤)
6059expr 448 . . . . . . . . . . . . . 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))
61 0elpw 4994 . . . . . . . . . . . . . . . . . . . . 21 ∅ ∈ 𝒫 𝑎
62 0fin 8397 . . . . . . . . . . . . . . . . . . . . 21 ∅ ∈ Fin
63 elin 3960 . . . . . . . . . . . . . . . . . . . . 21 (∅ ∈ (𝒫 𝑎 ∩ Fin) ↔ (∅ ∈ 𝒫 𝑎 ∧ ∅ ∈ Fin))
6461, 62, 63mpbir2an 702 . . . . . . . . . . . . . . . . . . . 20 ∅ ∈ (𝒫 𝑎 ∩ Fin)
65 unieq 4604 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = ∅ → 𝑏 = ∅)
6665eqeq2d 2775 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = ∅ → (𝑋 = 𝑏𝑋 = ∅))
6766notbid 309 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 = ∅ → (¬ 𝑋 = 𝑏 ↔ ¬ 𝑋 = ∅))
6867rspccv 3459 . . . . . . . . . . . . . . . . . . . 20 (∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏 → (∅ ∈ (𝒫 𝑎 ∩ Fin) → ¬ 𝑋 = ∅))
6964, 68mpi 20 . . . . . . . . . . . . . . . . . . 19 (∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏 → ¬ 𝑋 = ∅)
70 vex 3353 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑛 ∈ V
7170elpw 4323 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 ∈ 𝒫 𝑤𝑛𝑤)
72 elin 3960 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 ∈ (𝒫 𝑤 ∩ Fin) ↔ (𝑛 ∈ 𝒫 𝑤𝑛 ∈ Fin))
73 unieq 4604 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑏 = 𝑛 𝑏 = 𝑛)
7473eqeq2d 2775 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑏 = 𝑛 → (𝑋 = 𝑏𝑋 = 𝑛))
7574notbid 309 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑏 = 𝑛 → (¬ 𝑋 = 𝑏 ↔ ¬ 𝑋 = 𝑛))
7675rspccv 3459 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏 → (𝑛 ∈ (𝒫 𝑤 ∩ Fin) → ¬ 𝑋 = 𝑛))
7772, 76syl5bir 234 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏 → ((𝑛 ∈ 𝒫 𝑤𝑛 ∈ Fin) → ¬ 𝑋 = 𝑛))
7877expd 404 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏 → (𝑛 ∈ 𝒫 𝑤 → (𝑛 ∈ Fin → ¬ 𝑋 = 𝑛)))
7971, 78syl5bir 234 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏 → (𝑛𝑤 → (𝑛 ∈ Fin → ¬ 𝑋 = 𝑛)))
8079com23 86 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏 → (𝑛 ∈ Fin → (𝑛𝑤 → ¬ 𝑋 = 𝑛)))
8180ad2antll 720 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏)) → (𝑛 ∈ Fin → (𝑛𝑤 → ¬ 𝑋 = 𝑛)))
8281a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 𝑋 = ∅ → ((𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏)) → (𝑛 ∈ Fin → (𝑛𝑤 → ¬ 𝑋 = 𝑛))))
83 sseq2 3789 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑤 = ∅ → (𝑛𝑤𝑛 ⊆ ∅))
84 ss0 4138 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 ⊆ ∅ → 𝑛 = ∅)
8583, 84syl6bi 244 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = ∅ → (𝑛𝑤𝑛 = ∅))
86 unieq 4604 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 = ∅ → 𝑛 = ∅)
8786eqeq2d 2775 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 = ∅ → (𝑋 = 𝑛𝑋 = ∅))
8887notbid 309 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 = ∅ → (¬ 𝑋 = 𝑛 ↔ ¬ 𝑋 = ∅))
8988biimprcd 241 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑋 = ∅ → (𝑛 = ∅ → ¬ 𝑋 = 𝑛))
9089a1dd 50 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑋 = ∅ → (𝑛 = ∅ → (𝑛 ∈ Fin → ¬ 𝑋 = 𝑛)))
9185, 90syl9r 78 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑋 = ∅ → (𝑤 = ∅ → (𝑛𝑤 → (𝑛 ∈ Fin → ¬ 𝑋 = 𝑛))))
9291com34 91 . . . . . . . . . . . . . . . . . . . . . . 23 𝑋 = ∅ → (𝑤 = ∅ → (𝑛 ∈ Fin → (𝑛𝑤 → ¬ 𝑋 = 𝑛))))
9382, 92jaod 885 . . . . . . . . . . . . . . . . . . . . . 22 𝑋 = ∅ → (((𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = 𝑏)) ∨ 𝑤 = ∅) → (𝑛 ∈ Fin → (𝑛𝑤 → ¬ 𝑋 = 𝑛))))
9411, 93syl5bi 233 . . . . . . . . . . . . . . . . . . . . 21 𝑋 = ∅ → (𝑤 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) → (𝑛 ∈ Fin → (𝑛𝑤 → ¬ 𝑋 = 𝑛))))
951, 94sylan9r 504 . . . . . . . . . . . . . . . . . . . 20 ((¬ 𝑋 = ∅ ∧ 𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅})) → (𝑤𝑦 → (𝑛 ∈ Fin → (𝑛𝑤 → ¬ 𝑋 = 𝑛))))
9695com23 86 . . . . . . . . . . . . . . . . . . 19 ((¬ 𝑋 = ∅ ∧ 𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅})) → (𝑛 ∈ Fin → (𝑤𝑦 → (𝑛𝑤 → ¬ 𝑋 = 𝑛))))
9769, 96sylan 575 . . . . . . . . . . . . . . . . . 18 ((∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅})) → (𝑛 ∈ Fin → (𝑤𝑦 → (𝑛𝑤 → ¬ 𝑋 = 𝑛))))
9897ad2ant2lr 754 . . . . . . . . . . . . . . . . 17 ((((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))))
9998imp 395 . . . . . . . . . . . . . . . 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)))
10099adantrl 707 . . . . . . . . . . . . . . 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)) → (𝑤𝑦 → (𝑛𝑤 → ¬ 𝑋 = 𝑛)))
101100rexlimdv 3177 . . . . . . . . . . . . . 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))
10260, 101syld 47 . . . . . . . . . . . . 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 → ¬ 𝑋 = 𝑛))
103102expr 448 . . . . . . . . . . . 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)))
104103com23 86 . . . . . . . . . . 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 𝑦 → (𝑛 ∈ Fin → ¬ 𝑋 = 𝑛)))
105104impd 398 . . . . . . . . . 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))
10648, 105syl5bi 233 . . . . . . . . 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))
107106ralrimiv 3112 . . . . . . . 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)
108 unieq 4604 . . . . . . . . . . 11 (𝑛 = 𝑏 𝑛 = 𝑏)
109108eqeq2d 2775 . . . . . . . . . 10 (𝑛 = 𝑏 → (𝑋 = 𝑛𝑋 = 𝑏))
110109notbid 309 . . . . . . . . 9 (𝑛 = 𝑏 → (¬ 𝑋 = 𝑛 ↔ ¬ 𝑋 = 𝑏))
111110cbvralv 3319 . . . . . . . 8 (∀𝑛 ∈ (𝒫 𝑦 ∩ Fin) ¬ 𝑋 = 𝑛 ↔ ∀𝑏 ∈ (𝒫 𝑦 ∩ Fin) ¬ 𝑋 = 𝑏)
112107, 111sylib 209 . . . . . . 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)
11327, 47, 112jca32 511 . . . . . 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)))
114113ex 401 . . . . 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))))
115 orcom 896 . . . . . 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 ∈ {∅}))
11625elsn 4351 . . . . . . . 8 ( 𝑦 ∈ {∅} ↔ 𝑦 = ∅)
117 sseq2 3789 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝑎𝑧𝑎 𝑦))
118 pweq 4320 . . . . . . . . . . . 12 (𝑧 = 𝑦 → 𝒫 𝑧 = 𝒫 𝑦)
119118ineq1d 3977 . . . . . . . . . . 11 (𝑧 = 𝑦 → (𝒫 𝑧 ∩ Fin) = (𝒫 𝑦 ∩ Fin))
120119raleqdv 3292 . . . . . . . . . 10 (𝑧 = 𝑦 → (∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏 ↔ ∀𝑏 ∈ (𝒫 𝑦 ∩ Fin) ¬ 𝑋 = 𝑏))
121117, 120anbi12d 624 . . . . . . . . 9 (𝑧 = 𝑦 → ((𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏) ↔ (𝑎 𝑦 ∧ ∀𝑏 ∈ (𝒫 𝑦 ∩ Fin) ¬ 𝑋 = 𝑏)))
122121elrab 3521 . . . . . . . 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)))
123116, 122orbi12i 938 . . . . . . 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))))
124 df-or 874 . . . . . . 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))))
125123, 124bitr2i 267 . . . . . 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)}))
126 elun 3917 . . . . . 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 ∈ {∅}))
127115, 125, 1263bitr4i 294 . . . . 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)} ∪ {∅}))
128114, 127sylib 209 . . . 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)} ∪ {∅}))
129128ex 401 . . 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)} ∪ {∅})))
130129alrimiv 2022 . 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)} ∪ {∅})))
131 fvex 6390 . . . . . 6 (fi‘𝑥) ∈ V
132131pwex 5018 . . . . 5 𝒫 (fi‘𝑥) ∈ V
133132rabex 4975 . . . 4 {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∈ V
134 p0ex 5021 . . . 4 {∅} ∈ V
135133, 134unex 7156 . . 3 ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ∈ V
136135zorn 9584 . 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)
137130, 136syl 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 384  wo 873  w3a 1107  wal 1650   = wceq 1652  wex 1874  wcel 2155  wne 2937  wral 3055  wrex 3056  {crab 3059  cun 3732  cin 3733  wss 3734  wpss 3735  c0 4081  𝒫 cpw 4317  {csn 4336   cuni 4596   Or wor 5199  cfv 6070   [] crpss 7136  Fincfn 8162  ficfi 8525  topGenctg 16367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149  ax-ac2 9540
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6805  df-rpss 7137  df-om 7266  df-wrecs 7612  df-recs 7674  df-1o 7766  df-er 7949  df-en 8163  df-fin 8166  df-card 9018  df-ac 9192
This theorem is referenced by:  alexsubALTlem4  22136
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