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Theorem topdifinffinlem 33531
Description: This is the core of the proof of topdifinffin 33532, but to avoid the distinct variables on the definition, we need to split this proof into two. (Contributed by ML, 17-Jul-2020.)
Hypothesis
Ref Expression
topdifinf.t 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}
Assertion
Ref Expression
topdifinffinlem (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ Fin)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑇

Proof of Theorem topdifinffinlem
Dummy variables 𝑢 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1995 . . . . 5 𝑢 ¬ 𝐴 ∈ Fin
2 nfab1 2915 . . . . 5 𝑢{𝑢 ∣ ∃𝑦𝐴 𝑢 = {𝑦}}
3 nfcv 2913 . . . . 5 𝑢𝑇
4 abid 2759 . . . . . . . . . . 11 (𝑢 ∈ {𝑢 ∣ ∃𝑦𝐴 𝑢 = {𝑦}} ↔ ∃𝑦𝐴 𝑢 = {𝑦})
5 df-rex 3067 . . . . . . . . . . 11 (∃𝑦𝐴 𝑢 = {𝑦} ↔ ∃𝑦(𝑦𝐴𝑢 = {𝑦}))
64, 5bitri 264 . . . . . . . . . 10 (𝑢 ∈ {𝑢 ∣ ∃𝑦𝐴 𝑢 = {𝑦}} ↔ ∃𝑦(𝑦𝐴𝑢 = {𝑦}))
7 eqid 2771 . . . . . . . . . . . . . . 15 {𝑦} = {𝑦}
8 snex 5037 . . . . . . . . . . . . . . . . . 18 {𝑦} ∈ V
9 snelpwi 5041 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦𝐴 → {𝑦} ∈ 𝒫 𝐴)
10 eleq1 2838 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 = {𝑦} → (𝑥 ∈ 𝒫 𝐴 ↔ {𝑦} ∈ 𝒫 𝐴))
119, 10syl5ibr 236 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = {𝑦} → (𝑦𝐴𝑥 ∈ 𝒫 𝐴))
1211imdistani 558 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 = {𝑦} ∧ 𝑦𝐴) → (𝑥 = {𝑦} ∧ 𝑥 ∈ 𝒫 𝐴))
1312anim2i 603 . . . . . . . . . . . . . . . . . . . . . . . 24 ((¬ 𝐴 ∈ Fin ∧ (𝑥 = {𝑦} ∧ 𝑦𝐴)) → (¬ 𝐴 ∈ Fin ∧ (𝑥 = {𝑦} ∧ 𝑥 ∈ 𝒫 𝐴)))
14133impb 1107 . . . . . . . . . . . . . . . . . . . . . . 23 ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦𝐴) → (¬ 𝐴 ∈ Fin ∧ (𝑥 = {𝑦} ∧ 𝑥 ∈ 𝒫 𝐴)))
15 3anass 1080 . . . . . . . . . . . . . . . . . . . . . . 23 ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑥 ∈ 𝒫 𝐴) ↔ (¬ 𝐴 ∈ Fin ∧ (𝑥 = {𝑦} ∧ 𝑥 ∈ 𝒫 𝐴)))
1614, 15sylibr 224 . . . . . . . . . . . . . . . . . . . . . 22 ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦𝐴) → (¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑥 ∈ 𝒫 𝐴))
17 snfi 8197 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 {𝑦} ∈ Fin
18 eleq1 2838 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 = {𝑦} → (𝑥 ∈ Fin ↔ {𝑦} ∈ Fin))
1917, 18mpbiri 248 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 = {𝑦} → 𝑥 ∈ Fin)
20 difinf 8389 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((¬ 𝐴 ∈ Fin ∧ 𝑥 ∈ Fin) → ¬ (𝐴𝑥) ∈ Fin)
2119, 20sylan2 580 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦}) → ¬ (𝐴𝑥) ∈ Fin)
2221orcd 862 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦}) → (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)))
2322anim2i 603 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ 𝒫 𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦})) → (𝑥 ∈ 𝒫 𝐴 ∧ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
2423ancoms 446 . . . . . . . . . . . . . . . . . . . . . . 23 (((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦}) ∧ 𝑥 ∈ 𝒫 𝐴) → (𝑥 ∈ 𝒫 𝐴 ∧ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
25243impa 1100 . . . . . . . . . . . . . . . . . . . . . 22 ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑥 ∈ 𝒫 𝐴) → (𝑥 ∈ 𝒫 𝐴 ∧ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
2616, 25syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦𝐴) → (𝑥 ∈ 𝒫 𝐴 ∧ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
27 topdifinf.t . . . . . . . . . . . . . . . . . . . . . 22 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}
2827rabeq2i 3347 . . . . . . . . . . . . . . . . . . . . 21 (𝑥𝑇 ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
2926, 28sylibr 224 . . . . . . . . . . . . . . . . . . . 20 ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦𝐴) → 𝑥𝑇)
30 eleq1 2838 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = {𝑦} → (𝑥𝑇 ↔ {𝑦} ∈ 𝑇))
31303ad2ant2 1128 . . . . . . . . . . . . . . . . . . . 20 ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦𝐴) → (𝑥𝑇 ↔ {𝑦} ∈ 𝑇))
3229, 31mpbid 222 . . . . . . . . . . . . . . . . . . 19 ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦𝐴) → {𝑦} ∈ 𝑇)
3332sbcth 3602 . . . . . . . . . . . . . . . . . 18 ({𝑦} ∈ V → [{𝑦} / 𝑥]((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦𝐴) → {𝑦} ∈ 𝑇))
348, 33ax-mp 5 . . . . . . . . . . . . . . . . 17 [{𝑦} / 𝑥]((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦𝐴) → {𝑦} ∈ 𝑇)
35 sbcimg 3629 . . . . . . . . . . . . . . . . . 18 ({𝑦} ∈ V → ([{𝑦} / 𝑥]((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦𝐴) → {𝑦} ∈ 𝑇) ↔ ([{𝑦} / 𝑥]𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦𝐴) → [{𝑦} / 𝑥]{𝑦} ∈ 𝑇)))
368, 35ax-mp 5 . . . . . . . . . . . . . . . . 17 ([{𝑦} / 𝑥]((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦𝐴) → {𝑦} ∈ 𝑇) ↔ ([{𝑦} / 𝑥]𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦𝐴) → [{𝑦} / 𝑥]{𝑦} ∈ 𝑇))
3734, 36mpbi 220 . . . . . . . . . . . . . . . 16 ([{𝑦} / 𝑥]𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦𝐴) → [{𝑦} / 𝑥]{𝑦} ∈ 𝑇)
38 sbc3an 3645 . . . . . . . . . . . . . . . . . 18 ([{𝑦} / 𝑥]𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦𝐴) ↔ ([{𝑦} / 𝑥] ¬ 𝐴 ∈ Fin ∧ [{𝑦} / 𝑥]𝑥 = {𝑦} ∧ [{𝑦} / 𝑥]𝑦𝐴))
39 sbcg 3653 . . . . . . . . . . . . . . . . . . . 20 ({𝑦} ∈ V → ([{𝑦} / 𝑥] ¬ 𝐴 ∈ Fin ↔ ¬ 𝐴 ∈ Fin))
408, 39ax-mp 5 . . . . . . . . . . . . . . . . . . 19 ([{𝑦} / 𝑥] ¬ 𝐴 ∈ Fin ↔ ¬ 𝐴 ∈ Fin)
41403anbi1i 1160 . . . . . . . . . . . . . . . . . 18 (([{𝑦} / 𝑥] ¬ 𝐴 ∈ Fin ∧ [{𝑦} / 𝑥]𝑥 = {𝑦} ∧ [{𝑦} / 𝑥]𝑦𝐴) ↔ (¬ 𝐴 ∈ Fin ∧ [{𝑦} / 𝑥]𝑥 = {𝑦} ∧ [{𝑦} / 𝑥]𝑦𝐴))
42 eqsbc3 3627 . . . . . . . . . . . . . . . . . . . 20 ({𝑦} ∈ V → ([{𝑦} / 𝑥]𝑥 = {𝑦} ↔ {𝑦} = {𝑦}))
438, 42ax-mp 5 . . . . . . . . . . . . . . . . . . 19 ([{𝑦} / 𝑥]𝑥 = {𝑦} ↔ {𝑦} = {𝑦})
44433anbi2i 1161 . . . . . . . . . . . . . . . . . 18 ((¬ 𝐴 ∈ Fin ∧ [{𝑦} / 𝑥]𝑥 = {𝑦} ∧ [{𝑦} / 𝑥]𝑦𝐴) ↔ (¬ 𝐴 ∈ Fin ∧ {𝑦} = {𝑦} ∧ [{𝑦} / 𝑥]𝑦𝐴))
4538, 41, 443bitri 286 . . . . . . . . . . . . . . . . 17 ([{𝑦} / 𝑥]𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦𝐴) ↔ (¬ 𝐴 ∈ Fin ∧ {𝑦} = {𝑦} ∧ [{𝑦} / 𝑥]𝑦𝐴))
46 sbcg 3653 . . . . . . . . . . . . . . . . . . 19 ({𝑦} ∈ V → ([{𝑦} / 𝑥]𝑦𝐴𝑦𝐴))
478, 46ax-mp 5 . . . . . . . . . . . . . . . . . 18 ([{𝑦} / 𝑥]𝑦𝐴𝑦𝐴)
48473anbi3i 1162 . . . . . . . . . . . . . . . . 17 ((¬ 𝐴 ∈ Fin ∧ {𝑦} = {𝑦} ∧ [{𝑦} / 𝑥]𝑦𝐴) ↔ (¬ 𝐴 ∈ Fin ∧ {𝑦} = {𝑦} ∧ 𝑦𝐴))
4945, 48bitri 264 . . . . . . . . . . . . . . . 16 ([{𝑦} / 𝑥]𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦𝐴) ↔ (¬ 𝐴 ∈ Fin ∧ {𝑦} = {𝑦} ∧ 𝑦𝐴))
50 sbcg 3653 . . . . . . . . . . . . . . . . 17 ({𝑦} ∈ V → ([{𝑦} / 𝑥]{𝑦} ∈ 𝑇 ↔ {𝑦} ∈ 𝑇))
518, 50ax-mp 5 . . . . . . . . . . . . . . . 16 ([{𝑦} / 𝑥]{𝑦} ∈ 𝑇 ↔ {𝑦} ∈ 𝑇)
5237, 49, 513imtr3i 280 . . . . . . . . . . . . . . 15 ((¬ 𝐴 ∈ Fin ∧ {𝑦} = {𝑦} ∧ 𝑦𝐴) → {𝑦} ∈ 𝑇)
537, 52mp3an2 1560 . . . . . . . . . . . . . 14 ((¬ 𝐴 ∈ Fin ∧ 𝑦𝐴) → {𝑦} ∈ 𝑇)
5453ex 397 . . . . . . . . . . . . 13 𝐴 ∈ Fin → (𝑦𝐴 → {𝑦} ∈ 𝑇))
5554pm4.71d 551 . . . . . . . . . . . 12 𝐴 ∈ Fin → (𝑦𝐴 ↔ (𝑦𝐴 ∧ {𝑦} ∈ 𝑇)))
5655anbi1d 615 . . . . . . . . . . 11 𝐴 ∈ Fin → ((𝑦𝐴𝑢 = {𝑦}) ↔ ((𝑦𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑢 = {𝑦})))
5756exbidv 2002 . . . . . . . . . 10 𝐴 ∈ Fin → (∃𝑦(𝑦𝐴𝑢 = {𝑦}) ↔ ∃𝑦((𝑦𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑢 = {𝑦})))
586, 57syl5bb 272 . . . . . . . . 9 𝐴 ∈ Fin → (𝑢 ∈ {𝑢 ∣ ∃𝑦𝐴 𝑢 = {𝑦}} ↔ ∃𝑦((𝑦𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑢 = {𝑦})))
59 anass 454 . . . . . . . . . . 11 (((𝑦𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑢 = {𝑦}) ↔ (𝑦𝐴 ∧ ({𝑦} ∈ 𝑇𝑢 = {𝑦})))
6059exbii 1924 . . . . . . . . . 10 (∃𝑦((𝑦𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑢 = {𝑦}) ↔ ∃𝑦(𝑦𝐴 ∧ ({𝑦} ∈ 𝑇𝑢 = {𝑦})))
61 exsimpr 1947 . . . . . . . . . 10 (∃𝑦(𝑦𝐴 ∧ ({𝑦} ∈ 𝑇𝑢 = {𝑦})) → ∃𝑦({𝑦} ∈ 𝑇𝑢 = {𝑦}))
6260, 61sylbi 207 . . . . . . . . 9 (∃𝑦((𝑦𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑢 = {𝑦}) → ∃𝑦({𝑦} ∈ 𝑇𝑢 = {𝑦}))
6358, 62syl6bi 243 . . . . . . . 8 𝐴 ∈ Fin → (𝑢 ∈ {𝑢 ∣ ∃𝑦𝐴 𝑢 = {𝑦}} → ∃𝑦({𝑦} ∈ 𝑇𝑢 = {𝑦})))
64 ancom 448 . . . . . . . . . 10 (({𝑦} ∈ 𝑇𝑢 = {𝑦}) ↔ (𝑢 = {𝑦} ∧ {𝑦} ∈ 𝑇))
65 eleq1 2838 . . . . . . . . . . 11 (𝑢 = {𝑦} → (𝑢𝑇 ↔ {𝑦} ∈ 𝑇))
6665pm5.32i 564 . . . . . . . . . 10 ((𝑢 = {𝑦} ∧ 𝑢𝑇) ↔ (𝑢 = {𝑦} ∧ {𝑦} ∈ 𝑇))
6764, 66bitr4i 267 . . . . . . . . 9 (({𝑦} ∈ 𝑇𝑢 = {𝑦}) ↔ (𝑢 = {𝑦} ∧ 𝑢𝑇))
6867exbii 1924 . . . . . . . 8 (∃𝑦({𝑦} ∈ 𝑇𝑢 = {𝑦}) ↔ ∃𝑦(𝑢 = {𝑦} ∧ 𝑢𝑇))
6963, 68syl6ib 241 . . . . . . 7 𝐴 ∈ Fin → (𝑢 ∈ {𝑢 ∣ ∃𝑦𝐴 𝑢 = {𝑦}} → ∃𝑦(𝑢 = {𝑦} ∧ 𝑢𝑇)))
70 exsimpr 1947 . . . . . . 7 (∃𝑦(𝑢 = {𝑦} ∧ 𝑢𝑇) → ∃𝑦 𝑢𝑇)
7169, 70syl6 35 . . . . . 6 𝐴 ∈ Fin → (𝑢 ∈ {𝑢 ∣ ∃𝑦𝐴 𝑢 = {𝑦}} → ∃𝑦 𝑢𝑇))
72 ax5e 1993 . . . . . 6 (∃𝑦 𝑢𝑇𝑢𝑇)
7371, 72syl6 35 . . . . 5 𝐴 ∈ Fin → (𝑢 ∈ {𝑢 ∣ ∃𝑦𝐴 𝑢 = {𝑦}} → 𝑢𝑇))
741, 2, 3, 73ssrd 3757 . . . 4 𝐴 ∈ Fin → {𝑢 ∣ ∃𝑦𝐴 𝑢 = {𝑦}} ⊆ 𝑇)
75 eqid 2771 . . . . 5 {𝑢 ∣ ∃𝑦𝐴 𝑢 = {𝑦}} = {𝑢 ∣ ∃𝑦𝐴 𝑢 = {𝑦}}
7675dissneq 33524 . . . 4 (({𝑢 ∣ ∃𝑦𝐴 𝑢 = {𝑦}} ⊆ 𝑇𝑇 ∈ (TopOn‘𝐴)) → 𝑇 = 𝒫 𝐴)
7774, 76sylan 569 . . 3 ((¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴)) → 𝑇 = 𝒫 𝐴)
78 nfielex 8348 . . . . 5 𝐴 ∈ Fin → ∃𝑦 𝑦𝐴)
7978adantr 466 . . . 4 ((¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴)) → ∃𝑦 𝑦𝐴)
80 difss 3888 . . . . . . 7 (𝐴 ∖ {𝑦}) ⊆ 𝐴
81 elfvex 6364 . . . . . . . 8 (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ V)
82 difexg 4943 . . . . . . . 8 (𝐴 ∈ V → (𝐴 ∖ {𝑦}) ∈ V)
83 elpwg 4306 . . . . . . . 8 ((𝐴 ∖ {𝑦}) ∈ V → ((𝐴 ∖ {𝑦}) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ {𝑦}) ⊆ 𝐴))
8481, 82, 833syl 18 . . . . . . 7 (𝑇 ∈ (TopOn‘𝐴) → ((𝐴 ∖ {𝑦}) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ {𝑦}) ⊆ 𝐴))
8580, 84mpbiri 248 . . . . . 6 (𝑇 ∈ (TopOn‘𝐴) → (𝐴 ∖ {𝑦}) ∈ 𝒫 𝐴)
8685adantl 467 . . . . 5 ((¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴)) → (𝐴 ∖ {𝑦}) ∈ 𝒫 𝐴)
87 difinf 8389 . . . . . . . . . . . 12 ((¬ 𝐴 ∈ Fin ∧ {𝑦} ∈ Fin) → ¬ (𝐴 ∖ {𝑦}) ∈ Fin)
8817, 87mpan2 671 . . . . . . . . . . 11 𝐴 ∈ Fin → ¬ (𝐴 ∖ {𝑦}) ∈ Fin)
89 0fin 8347 . . . . . . . . . . . 12 ∅ ∈ Fin
90 eleq1 2838 . . . . . . . . . . . 12 ((𝐴 ∖ {𝑦}) = ∅ → ((𝐴 ∖ {𝑦}) ∈ Fin ↔ ∅ ∈ Fin))
9189, 90mpbiri 248 . . . . . . . . . . 11 ((𝐴 ∖ {𝑦}) = ∅ → (𝐴 ∖ {𝑦}) ∈ Fin)
9288, 91nsyl 137 . . . . . . . . . 10 𝐴 ∈ Fin → ¬ (𝐴 ∖ {𝑦}) = ∅)
9392ad2antrl 707 . . . . . . . . 9 ((𝑦𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴))) → ¬ (𝐴 ∖ {𝑦}) = ∅)
94 vsnid 4349 . . . . . . . . . . . . . 14 𝑦 ∈ {𝑦}
95 inelcm 4176 . . . . . . . . . . . . . 14 ((𝑦𝐴𝑦 ∈ {𝑦}) → (𝐴 ∩ {𝑦}) ≠ ∅)
9694, 95mpan2 671 . . . . . . . . . . . . 13 (𝑦𝐴 → (𝐴 ∩ {𝑦}) ≠ ∅)
97 disj4 4170 . . . . . . . . . . . . . 14 ((𝐴 ∩ {𝑦}) = ∅ ↔ ¬ (𝐴 ∖ {𝑦}) ⊊ 𝐴)
9897necon2abii 2993 . . . . . . . . . . . . 13 ((𝐴 ∖ {𝑦}) ⊊ 𝐴 ↔ (𝐴 ∩ {𝑦}) ≠ ∅)
9996, 98sylibr 224 . . . . . . . . . . . 12 (𝑦𝐴 → (𝐴 ∖ {𝑦}) ⊊ 𝐴)
10099pssned 3855 . . . . . . . . . . 11 (𝑦𝐴 → (𝐴 ∖ {𝑦}) ≠ 𝐴)
101100adantr 466 . . . . . . . . . 10 ((𝑦𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴))) → (𝐴 ∖ {𝑦}) ≠ 𝐴)
102101neneqd 2948 . . . . . . . . 9 ((𝑦𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴))) → ¬ (𝐴 ∖ {𝑦}) = 𝐴)
10393, 102jca 501 . . . . . . . 8 ((𝑦𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴))) → (¬ (𝐴 ∖ {𝑦}) = ∅ ∧ ¬ (𝐴 ∖ {𝑦}) = 𝐴))
104 pm4.56 973 . . . . . . . 8 ((¬ (𝐴 ∖ {𝑦}) = ∅ ∧ ¬ (𝐴 ∖ {𝑦}) = 𝐴) ↔ ¬ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴))
105103, 104sylib 208 . . . . . . 7 ((𝑦𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴))) → ¬ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴))
10685biantrurd 522 . . . . . . . . . 10 (𝑇 ∈ (TopOn‘𝐴) → ((¬ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin ∨ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴)) ↔ ((𝐴 ∖ {𝑦}) ∈ 𝒫 𝐴 ∧ (¬ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin ∨ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴)))))
107 difeq2 3873 . . . . . . . . . . . . . 14 (𝑥 = (𝐴 ∖ {𝑦}) → (𝐴𝑥) = (𝐴 ∖ (𝐴 ∖ {𝑦})))
108107eleq1d 2835 . . . . . . . . . . . . 13 (𝑥 = (𝐴 ∖ {𝑦}) → ((𝐴𝑥) ∈ Fin ↔ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin))
109108notbid 307 . . . . . . . . . . . 12 (𝑥 = (𝐴 ∖ {𝑦}) → (¬ (𝐴𝑥) ∈ Fin ↔ ¬ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin))
110 eqeq1 2775 . . . . . . . . . . . . 13 (𝑥 = (𝐴 ∖ {𝑦}) → (𝑥 = ∅ ↔ (𝐴 ∖ {𝑦}) = ∅))
111 eqeq1 2775 . . . . . . . . . . . . 13 (𝑥 = (𝐴 ∖ {𝑦}) → (𝑥 = 𝐴 ↔ (𝐴 ∖ {𝑦}) = 𝐴))
112110, 111orbi12d 904 . . . . . . . . . . . 12 (𝑥 = (𝐴 ∖ {𝑦}) → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴)))
113109, 112orbi12d 904 . . . . . . . . . . 11 (𝑥 = (𝐴 ∖ {𝑦}) → ((¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) ↔ (¬ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin ∨ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴))))
114113, 27elrab2 3518 . . . . . . . . . 10 ((𝐴 ∖ {𝑦}) ∈ 𝑇 ↔ ((𝐴 ∖ {𝑦}) ∈ 𝒫 𝐴 ∧ (¬ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin ∨ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴))))
115106, 114syl6rbbr 279 . . . . . . . . 9 (𝑇 ∈ (TopOn‘𝐴) → ((𝐴 ∖ {𝑦}) ∈ 𝑇 ↔ (¬ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin ∨ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴))))
116 dfin4 4016 . . . . . . . . . . 11 (𝐴 ∩ {𝑦}) = (𝐴 ∖ (𝐴 ∖ {𝑦}))
117 inss2 3982 . . . . . . . . . . . 12 (𝐴 ∩ {𝑦}) ⊆ {𝑦}
118 ssfi 8339 . . . . . . . . . . . 12 (({𝑦} ∈ Fin ∧ (𝐴 ∩ {𝑦}) ⊆ {𝑦}) → (𝐴 ∩ {𝑦}) ∈ Fin)
11917, 117, 118mp2an 672 . . . . . . . . . . 11 (𝐴 ∩ {𝑦}) ∈ Fin
120116, 119eqeltrri 2847 . . . . . . . . . 10 (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin
121 biortn 923 . . . . . . . . . 10 ((𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin → (((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴) ↔ (¬ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin ∨ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴))))
122120, 121ax-mp 5 . . . . . . . . 9 (((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴) ↔ (¬ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin ∨ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴)))
123115, 122syl6bbr 278 . . . . . . . 8 (𝑇 ∈ (TopOn‘𝐴) → ((𝐴 ∖ {𝑦}) ∈ 𝑇 ↔ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴)))
124123ad2antll 708 . . . . . . 7 ((𝑦𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴))) → ((𝐴 ∖ {𝑦}) ∈ 𝑇 ↔ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴)))
125105, 124mtbird 314 . . . . . 6 ((𝑦𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴))) → ¬ (𝐴 ∖ {𝑦}) ∈ 𝑇)
126125expcom 398 . . . . 5 ((¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴)) → (𝑦𝐴 → ¬ (𝐴 ∖ {𝑦}) ∈ 𝑇))
127 nelneq2 2875 . . . . . 6 (((𝐴 ∖ {𝑦}) ∈ 𝒫 𝐴 ∧ ¬ (𝐴 ∖ {𝑦}) ∈ 𝑇) → ¬ 𝒫 𝐴 = 𝑇)
128 eqcom 2778 . . . . . 6 (𝑇 = 𝒫 𝐴 ↔ 𝒫 𝐴 = 𝑇)
129127, 128sylnibr 318 . . . . 5 (((𝐴 ∖ {𝑦}) ∈ 𝒫 𝐴 ∧ ¬ (𝐴 ∖ {𝑦}) ∈ 𝑇) → ¬ 𝑇 = 𝒫 𝐴)
13086, 126, 129syl6an 663 . . . 4 ((¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴)) → (𝑦𝐴 → ¬ 𝑇 = 𝒫 𝐴))
13179, 130exellimddv 33529 . . 3 ((¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴)) → ¬ 𝑇 = 𝒫 𝐴)
13277, 131pm2.65da 818 . 2 𝐴 ∈ Fin → ¬ 𝑇 ∈ (TopOn‘𝐴))
133132con4i 114 1 (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 836  w3a 1071   = wceq 1631  wex 1852  wcel 2145  {cab 2757  wne 2943  wrex 3062  {crab 3065  Vcvv 3351  [wsbc 3587  cdif 3720  cin 3722  wss 3723  wpss 3724  c0 4063  𝒫 cpw 4298  {csn 4317  cfv 6030  Fincfn 8112  TopOnctopon 20934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7099
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6798  df-oprab 6799  df-mpt2 6800  df-om 7216  df-wrecs 7562  df-recs 7624  df-rdg 7662  df-1o 7716  df-oadd 7720  df-er 7899  df-en 8113  df-fin 8116  df-topgen 16311  df-top 20918  df-topon 20935
This theorem is referenced by:  topdifinffin  33532
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