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Theorem topdifinffinlem 33505
Description: This is the core of the proof of topdifinffin 33506, 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 2005 . . . . 5 𝑢 ¬ 𝐴 ∈ Fin
2 nfab1 2946 . . . . 5 𝑢{𝑢 ∣ ∃𝑦𝐴 𝑢 = {𝑦}}
3 nfcv 2944 . . . . 5 𝑢𝑇
4 abid 2790 . . . . . . . . . . 11 (𝑢 ∈ {𝑢 ∣ ∃𝑦𝐴 𝑢 = {𝑦}} ↔ ∃𝑦𝐴 𝑢 = {𝑦})
5 df-rex 3098 . . . . . . . . . . 11 (∃𝑦𝐴 𝑢 = {𝑦} ↔ ∃𝑦(𝑦𝐴𝑢 = {𝑦}))
64, 5bitri 266 . . . . . . . . . 10 (𝑢 ∈ {𝑢 ∣ ∃𝑦𝐴 𝑢 = {𝑦}} ↔ ∃𝑦(𝑦𝐴𝑢 = {𝑦}))
7 eqid 2802 . . . . . . . . . . . . . . 15 {𝑦} = {𝑦}
8 snex 5092 . . . . . . . . . . . . . . . . . 18 {𝑦} ∈ V
9 snelpwi 5096 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦𝐴 → {𝑦} ∈ 𝒫 𝐴)
10 eleq1 2869 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 = {𝑦} → (𝑥 ∈ 𝒫 𝐴 ↔ {𝑦} ∈ 𝒫 𝐴))
119, 10syl5ibr 237 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = {𝑦} → (𝑦𝐴𝑥 ∈ 𝒫 𝐴))
1211imdistani 560 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 = {𝑦} ∧ 𝑦𝐴) → (𝑥 = {𝑦} ∧ 𝑥 ∈ 𝒫 𝐴))
1312anim2i 605 . . . . . . . . . . . . . . . . . . . . . . . 24 ((¬ 𝐴 ∈ Fin ∧ (𝑥 = {𝑦} ∧ 𝑦𝐴)) → (¬ 𝐴 ∈ Fin ∧ (𝑥 = {𝑦} ∧ 𝑥 ∈ 𝒫 𝐴)))
14133impb 1136 . . . . . . . . . . . . . . . . . . . . . . 23 ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦𝐴) → (¬ 𝐴 ∈ Fin ∧ (𝑥 = {𝑦} ∧ 𝑥 ∈ 𝒫 𝐴)))
15 3anass 1109 . . . . . . . . . . . . . . . . . . . . . . 23 ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑥 ∈ 𝒫 𝐴) ↔ (¬ 𝐴 ∈ Fin ∧ (𝑥 = {𝑦} ∧ 𝑥 ∈ 𝒫 𝐴)))
1614, 15sylibr 225 . . . . . . . . . . . . . . . . . . . . . 22 ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦𝐴) → (¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑥 ∈ 𝒫 𝐴))
17 snfi 8271 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 {𝑦} ∈ Fin
18 eleq1 2869 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 = {𝑦} → (𝑥 ∈ Fin ↔ {𝑦} ∈ Fin))
1917, 18mpbiri 249 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 = {𝑦} → 𝑥 ∈ Fin)
20 difinf 8463 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((¬ 𝐴 ∈ Fin ∧ 𝑥 ∈ Fin) → ¬ (𝐴𝑥) ∈ Fin)
2119, 20sylan2 582 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦}) → ¬ (𝐴𝑥) ∈ Fin)
2221orcd 891 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦}) → (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)))
2322anim2i 605 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ 𝒫 𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦})) → (𝑥 ∈ 𝒫 𝐴 ∧ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
2423ancoms 448 . . . . . . . . . . . . . . . . . . . . . . 23 (((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦}) ∧ 𝑥 ∈ 𝒫 𝐴) → (𝑥 ∈ 𝒫 𝐴 ∧ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
25243impa 1129 . . . . . . . . . . . . . . . . . . . . . 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 3383 . . . . . . . . . . . . . . . . . . . . 21 (𝑥𝑇 ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
2926, 28sylibr 225 . . . . . . . . . . . . . . . . . . . 20 ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦𝐴) → 𝑥𝑇)
30 eleq1 2869 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = {𝑦} → (𝑥𝑇 ↔ {𝑦} ∈ 𝑇))
31303ad2ant2 1157 . . . . . . . . . . . . . . . . . . . 20 ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦𝐴) → (𝑥𝑇 ↔ {𝑦} ∈ 𝑇))
3229, 31mpbid 223 . . . . . . . . . . . . . . . . . . 19 ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦𝐴) → {𝑦} ∈ 𝑇)
3332sbcth 3642 . . . . . . . . . . . . . . . . . 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 3669 . . . . . . . . . . . . . . . . . 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 221 . . . . . . . . . . . . . . . 16 ([{𝑦} / 𝑥]𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦𝐴) → [{𝑦} / 𝑥]{𝑦} ∈ 𝑇)
38 sbc3an 3685 . . . . . . . . . . . . . . . . . 18 ([{𝑦} / 𝑥]𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦𝐴) ↔ ([{𝑦} / 𝑥] ¬ 𝐴 ∈ Fin ∧ [{𝑦} / 𝑥]𝑥 = {𝑦} ∧ [{𝑦} / 𝑥]𝑦𝐴))
39 sbcg 3693 . . . . . . . . . . . . . . . . . . . 20 ({𝑦} ∈ V → ([{𝑦} / 𝑥] ¬ 𝐴 ∈ Fin ↔ ¬ 𝐴 ∈ Fin))
408, 39ax-mp 5 . . . . . . . . . . . . . . . . . . 19 ([{𝑦} / 𝑥] ¬ 𝐴 ∈ Fin ↔ ¬ 𝐴 ∈ Fin)
41403anbi1i 1189 . . . . . . . . . . . . . . . . . 18 (([{𝑦} / 𝑥] ¬ 𝐴 ∈ Fin ∧ [{𝑦} / 𝑥]𝑥 = {𝑦} ∧ [{𝑦} / 𝑥]𝑦𝐴) ↔ (¬ 𝐴 ∈ Fin ∧ [{𝑦} / 𝑥]𝑥 = {𝑦} ∧ [{𝑦} / 𝑥]𝑦𝐴))
42 eqsbc3 3667 . . . . . . . . . . . . . . . . . . . 20 ({𝑦} ∈ V → ([{𝑦} / 𝑥]𝑥 = {𝑦} ↔ {𝑦} = {𝑦}))
438, 42ax-mp 5 . . . . . . . . . . . . . . . . . . 19 ([{𝑦} / 𝑥]𝑥 = {𝑦} ↔ {𝑦} = {𝑦})
44433anbi2i 1190 . . . . . . . . . . . . . . . . . 18 ((¬ 𝐴 ∈ Fin ∧ [{𝑦} / 𝑥]𝑥 = {𝑦} ∧ [{𝑦} / 𝑥]𝑦𝐴) ↔ (¬ 𝐴 ∈ Fin ∧ {𝑦} = {𝑦} ∧ [{𝑦} / 𝑥]𝑦𝐴))
4538, 41, 443bitri 288 . . . . . . . . . . . . . . . . 17 ([{𝑦} / 𝑥]𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦𝐴) ↔ (¬ 𝐴 ∈ Fin ∧ {𝑦} = {𝑦} ∧ [{𝑦} / 𝑥]𝑦𝐴))
46 sbcg 3693 . . . . . . . . . . . . . . . . . . 19 ({𝑦} ∈ V → ([{𝑦} / 𝑥]𝑦𝐴𝑦𝐴))
478, 46ax-mp 5 . . . . . . . . . . . . . . . . . 18 ([{𝑦} / 𝑥]𝑦𝐴𝑦𝐴)
48473anbi3i 1191 . . . . . . . . . . . . . . . . 17 ((¬ 𝐴 ∈ Fin ∧ {𝑦} = {𝑦} ∧ [{𝑦} / 𝑥]𝑦𝐴) ↔ (¬ 𝐴 ∈ Fin ∧ {𝑦} = {𝑦} ∧ 𝑦𝐴))
4945, 48bitri 266 . . . . . . . . . . . . . . . 16 ([{𝑦} / 𝑥]𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦𝐴) ↔ (¬ 𝐴 ∈ Fin ∧ {𝑦} = {𝑦} ∧ 𝑦𝐴))
50 sbcg 3693 . . . . . . . . . . . . . . . . 17 ({𝑦} ∈ V → ([{𝑦} / 𝑥]{𝑦} ∈ 𝑇 ↔ {𝑦} ∈ 𝑇))
518, 50ax-mp 5 . . . . . . . . . . . . . . . 16 ([{𝑦} / 𝑥]{𝑦} ∈ 𝑇 ↔ {𝑦} ∈ 𝑇)
5237, 49, 513imtr3i 282 . . . . . . . . . . . . . . 15 ((¬ 𝐴 ∈ Fin ∧ {𝑦} = {𝑦} ∧ 𝑦𝐴) → {𝑦} ∈ 𝑇)
537, 52mp3an2 1566 . . . . . . . . . . . . . 14 ((¬ 𝐴 ∈ Fin ∧ 𝑦𝐴) → {𝑦} ∈ 𝑇)
5453ex 399 . . . . . . . . . . . . 13 𝐴 ∈ Fin → (𝑦𝐴 → {𝑦} ∈ 𝑇))
5554pm4.71d 553 . . . . . . . . . . . 12 𝐴 ∈ Fin → (𝑦𝐴 ↔ (𝑦𝐴 ∧ {𝑦} ∈ 𝑇)))
5655anbi1d 617 . . . . . . . . . . 11 𝐴 ∈ Fin → ((𝑦𝐴𝑢 = {𝑦}) ↔ ((𝑦𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑢 = {𝑦})))
5756exbidv 2012 . . . . . . . . . 10 𝐴 ∈ Fin → (∃𝑦(𝑦𝐴𝑢 = {𝑦}) ↔ ∃𝑦((𝑦𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑢 = {𝑦})))
586, 57syl5bb 274 . . . . . . . . 9 𝐴 ∈ Fin → (𝑢 ∈ {𝑢 ∣ ∃𝑦𝐴 𝑢 = {𝑦}} ↔ ∃𝑦((𝑦𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑢 = {𝑦})))
59 anass 456 . . . . . . . . . . 11 (((𝑦𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑢 = {𝑦}) ↔ (𝑦𝐴 ∧ ({𝑦} ∈ 𝑇𝑢 = {𝑦})))
6059exbii 1933 . . . . . . . . . 10 (∃𝑦((𝑦𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑢 = {𝑦}) ↔ ∃𝑦(𝑦𝐴 ∧ ({𝑦} ∈ 𝑇𝑢 = {𝑦})))
61 exsimpr 1957 . . . . . . . . . 10 (∃𝑦(𝑦𝐴 ∧ ({𝑦} ∈ 𝑇𝑢 = {𝑦})) → ∃𝑦({𝑦} ∈ 𝑇𝑢 = {𝑦}))
6260, 61sylbi 208 . . . . . . . . 9 (∃𝑦((𝑦𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑢 = {𝑦}) → ∃𝑦({𝑦} ∈ 𝑇𝑢 = {𝑦}))
6358, 62syl6bi 244 . . . . . . . 8 𝐴 ∈ Fin → (𝑢 ∈ {𝑢 ∣ ∃𝑦𝐴 𝑢 = {𝑦}} → ∃𝑦({𝑦} ∈ 𝑇𝑢 = {𝑦})))
64 ancom 450 . . . . . . . . . 10 (({𝑦} ∈ 𝑇𝑢 = {𝑦}) ↔ (𝑢 = {𝑦} ∧ {𝑦} ∈ 𝑇))
65 eleq1 2869 . . . . . . . . . . 11 (𝑢 = {𝑦} → (𝑢𝑇 ↔ {𝑦} ∈ 𝑇))
6665pm5.32i 566 . . . . . . . . . 10 ((𝑢 = {𝑦} ∧ 𝑢𝑇) ↔ (𝑢 = {𝑦} ∧ {𝑦} ∈ 𝑇))
6764, 66bitr4i 269 . . . . . . . . 9 (({𝑦} ∈ 𝑇𝑢 = {𝑦}) ↔ (𝑢 = {𝑦} ∧ 𝑢𝑇))
6867exbii 1933 . . . . . . . 8 (∃𝑦({𝑦} ∈ 𝑇𝑢 = {𝑦}) ↔ ∃𝑦(𝑢 = {𝑦} ∧ 𝑢𝑇))
6963, 68syl6ib 242 . . . . . . 7 𝐴 ∈ Fin → (𝑢 ∈ {𝑢 ∣ ∃𝑦𝐴 𝑢 = {𝑦}} → ∃𝑦(𝑢 = {𝑦} ∧ 𝑢𝑇)))
70 exsimpr 1957 . . . . . . 7 (∃𝑦(𝑢 = {𝑦} ∧ 𝑢𝑇) → ∃𝑦 𝑢𝑇)
7169, 70syl6 35 . . . . . 6 𝐴 ∈ Fin → (𝑢 ∈ {𝑢 ∣ ∃𝑦𝐴 𝑢 = {𝑦}} → ∃𝑦 𝑢𝑇))
72 ax5e 2003 . . . . . 6 (∃𝑦 𝑢𝑇𝑢𝑇)
7371, 72syl6 35 . . . . 5 𝐴 ∈ Fin → (𝑢 ∈ {𝑢 ∣ ∃𝑦𝐴 𝑢 = {𝑦}} → 𝑢𝑇))
741, 2, 3, 73ssrd 3797 . . . 4 𝐴 ∈ Fin → {𝑢 ∣ ∃𝑦𝐴 𝑢 = {𝑦}} ⊆ 𝑇)
75 eqid 2802 . . . . 5 {𝑢 ∣ ∃𝑦𝐴 𝑢 = {𝑦}} = {𝑢 ∣ ∃𝑦𝐴 𝑢 = {𝑦}}
7675dissneq 33499 . . . 4 (({𝑢 ∣ ∃𝑦𝐴 𝑢 = {𝑦}} ⊆ 𝑇𝑇 ∈ (TopOn‘𝐴)) → 𝑇 = 𝒫 𝐴)
7774, 76sylan 571 . . 3 ((¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴)) → 𝑇 = 𝒫 𝐴)
78 nfielex 8422 . . . . 5 𝐴 ∈ Fin → ∃𝑦 𝑦𝐴)
7978adantr 468 . . . 4 ((¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴)) → ∃𝑦 𝑦𝐴)
80 difss 3930 . . . . . . 7 (𝐴 ∖ {𝑦}) ⊆ 𝐴
81 elfvex 6435 . . . . . . . 8 (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ V)
82 difexg 4997 . . . . . . . 8 (𝐴 ∈ V → (𝐴 ∖ {𝑦}) ∈ V)
83 elpwg 4353 . . . . . . . 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 249 . . . . . 6 (𝑇 ∈ (TopOn‘𝐴) → (𝐴 ∖ {𝑦}) ∈ 𝒫 𝐴)
8685adantl 469 . . . . 5 ((¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴)) → (𝐴 ∖ {𝑦}) ∈ 𝒫 𝐴)
87 difinf 8463 . . . . . . . . . . . 12 ((¬ 𝐴 ∈ Fin ∧ {𝑦} ∈ Fin) → ¬ (𝐴 ∖ {𝑦}) ∈ Fin)
8817, 87mpan2 674 . . . . . . . . . . 11 𝐴 ∈ Fin → ¬ (𝐴 ∖ {𝑦}) ∈ Fin)
89 0fin 8421 . . . . . . . . . . . 12 ∅ ∈ Fin
90 eleq1 2869 . . . . . . . . . . . 12 ((𝐴 ∖ {𝑦}) = ∅ → ((𝐴 ∖ {𝑦}) ∈ Fin ↔ ∅ ∈ Fin))
9189, 90mpbiri 249 . . . . . . . . . . 11 ((𝐴 ∖ {𝑦}) = ∅ → (𝐴 ∖ {𝑦}) ∈ Fin)
9288, 91nsyl 137 . . . . . . . . . 10 𝐴 ∈ Fin → ¬ (𝐴 ∖ {𝑦}) = ∅)
9392ad2antrl 710 . . . . . . . . 9 ((𝑦𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴))) → ¬ (𝐴 ∖ {𝑦}) = ∅)
94 vsnid 4397 . . . . . . . . . . . . . 14 𝑦 ∈ {𝑦}
95 inelcm 4223 . . . . . . . . . . . . . 14 ((𝑦𝐴𝑦 ∈ {𝑦}) → (𝐴 ∩ {𝑦}) ≠ ∅)
9694, 95mpan2 674 . . . . . . . . . . . . 13 (𝑦𝐴 → (𝐴 ∩ {𝑦}) ≠ ∅)
97 disj4 4217 . . . . . . . . . . . . . 14 ((𝐴 ∩ {𝑦}) = ∅ ↔ ¬ (𝐴 ∖ {𝑦}) ⊊ 𝐴)
9897necon2abii 3024 . . . . . . . . . . . . 13 ((𝐴 ∖ {𝑦}) ⊊ 𝐴 ↔ (𝐴 ∩ {𝑦}) ≠ ∅)
9996, 98sylibr 225 . . . . . . . . . . . 12 (𝑦𝐴 → (𝐴 ∖ {𝑦}) ⊊ 𝐴)
10099pssned 3897 . . . . . . . . . . 11 (𝑦𝐴 → (𝐴 ∖ {𝑦}) ≠ 𝐴)
101100adantr 468 . . . . . . . . . 10 ((𝑦𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴))) → (𝐴 ∖ {𝑦}) ≠ 𝐴)
102101neneqd 2979 . . . . . . . . 9 ((𝑦𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴))) → ¬ (𝐴 ∖ {𝑦}) = 𝐴)
10393, 102jca 503 . . . . . . . 8 ((𝑦𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴))) → (¬ (𝐴 ∖ {𝑦}) = ∅ ∧ ¬ (𝐴 ∖ {𝑦}) = 𝐴))
104 pm4.56 1002 . . . . . . . 8 ((¬ (𝐴 ∖ {𝑦}) = ∅ ∧ ¬ (𝐴 ∖ {𝑦}) = 𝐴) ↔ ¬ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴))
105103, 104sylib 209 . . . . . . 7 ((𝑦𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴))) → ¬ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴))
10685biantrurd 524 . . . . . . . . . 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 3915 . . . . . . . . . . . . . 14 (𝑥 = (𝐴 ∖ {𝑦}) → (𝐴𝑥) = (𝐴 ∖ (𝐴 ∖ {𝑦})))
108107eleq1d 2866 . . . . . . . . . . . . 13 (𝑥 = (𝐴 ∖ {𝑦}) → ((𝐴𝑥) ∈ Fin ↔ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin))
109108notbid 309 . . . . . . . . . . . 12 (𝑥 = (𝐴 ∖ {𝑦}) → (¬ (𝐴𝑥) ∈ Fin ↔ ¬ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin))
110 eqeq1 2806 . . . . . . . . . . . . 13 (𝑥 = (𝐴 ∖ {𝑦}) → (𝑥 = ∅ ↔ (𝐴 ∖ {𝑦}) = ∅))
111 eqeq1 2806 . . . . . . . . . . . . 13 (𝑥 = (𝐴 ∖ {𝑦}) → (𝑥 = 𝐴 ↔ (𝐴 ∖ {𝑦}) = 𝐴))
112110, 111orbi12d 933 . . . . . . . . . . . 12 (𝑥 = (𝐴 ∖ {𝑦}) → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴)))
113109, 112orbi12d 933 . . . . . . . . . . 11 (𝑥 = (𝐴 ∖ {𝑦}) → ((¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) ↔ (¬ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin ∨ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴))))
114113, 27elrab2 3558 . . . . . . . . . 10 ((𝐴 ∖ {𝑦}) ∈ 𝑇 ↔ ((𝐴 ∖ {𝑦}) ∈ 𝒫 𝐴 ∧ (¬ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin ∨ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴))))
115106, 114syl6rbbr 281 . . . . . . . . 9 (𝑇 ∈ (TopOn‘𝐴) → ((𝐴 ∖ {𝑦}) ∈ 𝑇 ↔ (¬ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin ∨ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴))))
116 dfin4 4063 . . . . . . . . . . 11 (𝐴 ∩ {𝑦}) = (𝐴 ∖ (𝐴 ∖ {𝑦}))
117 inss2 4024 . . . . . . . . . . . 12 (𝐴 ∩ {𝑦}) ⊆ {𝑦}
118 ssfi 8413 . . . . . . . . . . . 12 (({𝑦} ∈ Fin ∧ (𝐴 ∩ {𝑦}) ⊆ {𝑦}) → (𝐴 ∩ {𝑦}) ∈ Fin)
11917, 117, 118mp2an 675 . . . . . . . . . . 11 (𝐴 ∩ {𝑦}) ∈ Fin
120116, 119eqeltrri 2878 . . . . . . . . . 10 (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin
121 biortn 952 . . . . . . . . . 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 280 . . . . . . . 8 (𝑇 ∈ (TopOn‘𝐴) → ((𝐴 ∖ {𝑦}) ∈ 𝑇 ↔ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴)))
124123ad2antll 711 . . . . . . 7 ((𝑦𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴))) → ((𝐴 ∖ {𝑦}) ∈ 𝑇 ↔ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴)))
125105, 124mtbird 316 . . . . . 6 ((𝑦𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴))) → ¬ (𝐴 ∖ {𝑦}) ∈ 𝑇)
126125expcom 400 . . . . 5 ((¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴)) → (𝑦𝐴 → ¬ (𝐴 ∖ {𝑦}) ∈ 𝑇))
127 nelneq2 2906 . . . . . 6 (((𝐴 ∖ {𝑦}) ∈ 𝒫 𝐴 ∧ ¬ (𝐴 ∖ {𝑦}) ∈ 𝑇) → ¬ 𝒫 𝐴 = 𝑇)
128 eqcom 2809 . . . . . 6 (𝑇 = 𝒫 𝐴 ↔ 𝒫 𝐴 = 𝑇)
129127, 128sylnibr 320 . . . . 5 (((𝐴 ∖ {𝑦}) ∈ 𝒫 𝐴 ∧ ¬ (𝐴 ∖ {𝑦}) ∈ 𝑇) → ¬ 𝑇 = 𝒫 𝐴)
13086, 126, 129syl6an 666 . . . 4 ((¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴)) → (𝑦𝐴 → ¬ 𝑇 = 𝒫 𝐴))
13179, 130exellimddv 33503 . . 3 ((¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴)) → ¬ 𝑇 = 𝒫 𝐴)
13277, 131pm2.65da 842 . 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 197  wa 384  wo 865  w3a 1100   = wceq 1637  wex 1859  wcel 2155  {cab 2788  wne 2974  wrex 3093  {crab 3096  Vcvv 3387  [wsbc 3627  cdif 3760  cin 3762  wss 3763  wpss 3764  c0 4110  𝒫 cpw 4345  {csn 4364  cfv 6095  Fincfn 8186  TopOnctopon 20922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-ral 3097  df-rex 3098  df-reu 3099  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-pss 3779  df-nul 4111  df-if 4274  df-pw 4347  df-sn 4365  df-pr 4367  df-tp 4369  df-op 4371  df-uni 4624  df-int 4663  df-iun 4707  df-br 4838  df-opab 4900  df-mpt 4917  df-tr 4940  df-id 5213  df-eprel 5218  df-po 5226  df-so 5227  df-fr 5264  df-we 5266  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-pred 5887  df-ord 5933  df-on 5934  df-lim 5935  df-suc 5936  df-iota 6058  df-fun 6097  df-fn 6098  df-f 6099  df-f1 6100  df-fo 6101  df-f1o 6102  df-fv 6103  df-ov 6871  df-oprab 6872  df-mpt2 6873  df-om 7290  df-wrecs 7636  df-recs 7698  df-rdg 7736  df-1o 7790  df-oadd 7794  df-er 7973  df-en 8187  df-fin 8190  df-topgen 16303  df-top 20906  df-topon 20923
This theorem is referenced by:  topdifinffin  33506
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