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Theorem topdifinffinlem 35847
Description: This is the core of the proof of topdifinffin 35848, 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 = {π‘₯ ∈ 𝒫 𝐴 ∣ (Β¬ (𝐴 βˆ– π‘₯) ∈ Fin ∨ (π‘₯ = βˆ… ∨ π‘₯ = 𝐴))}
Assertion
Ref Expression
topdifinffinlem (𝑇 ∈ (TopOnβ€˜π΄) β†’ 𝐴 ∈ Fin)
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝑇

Proof of Theorem topdifinffinlem
Dummy variables 𝑒 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1918 . . . . 5 Ⅎ𝑒 Β¬ 𝐴 ∈ Fin
2 nfab1 2910 . . . . 5 Ⅎ𝑒{𝑒 ∣ βˆƒπ‘¦ ∈ 𝐴 𝑒 = {𝑦}}
3 nfcv 2908 . . . . 5 Ⅎ𝑒𝑇
4 abid 2718 . . . . . . . . . . 11 (𝑒 ∈ {𝑒 ∣ βˆƒπ‘¦ ∈ 𝐴 𝑒 = {𝑦}} ↔ βˆƒπ‘¦ ∈ 𝐴 𝑒 = {𝑦})
5 df-rex 3075 . . . . . . . . . . 11 (βˆƒπ‘¦ ∈ 𝐴 𝑒 = {𝑦} ↔ βˆƒπ‘¦(𝑦 ∈ 𝐴 ∧ 𝑒 = {𝑦}))
64, 5bitri 275 . . . . . . . . . 10 (𝑒 ∈ {𝑒 ∣ βˆƒπ‘¦ ∈ 𝐴 𝑒 = {𝑦}} ↔ βˆƒπ‘¦(𝑦 ∈ 𝐴 ∧ 𝑒 = {𝑦}))
7 eqid 2737 . . . . . . . . . . . . . . 15 {𝑦} = {𝑦}
8 vsnex 5391 . . . . . . . . . . . . . . . . . 18 {𝑦} ∈ V
9 snelpwi 5405 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ 𝐴 β†’ {𝑦} ∈ 𝒫 𝐴)
10 eleq1 2826 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (π‘₯ = {𝑦} β†’ (π‘₯ ∈ 𝒫 𝐴 ↔ {𝑦} ∈ 𝒫 𝐴))
119, 10syl5ibr 246 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (π‘₯ = {𝑦} β†’ (𝑦 ∈ 𝐴 β†’ π‘₯ ∈ 𝒫 𝐴))
1211imdistani 570 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((π‘₯ = {𝑦} ∧ 𝑦 ∈ 𝐴) β†’ (π‘₯ = {𝑦} ∧ π‘₯ ∈ 𝒫 𝐴))
1312anim2i 618 . . . . . . . . . . . . . . . . . . . . . . . 24 ((Β¬ 𝐴 ∈ Fin ∧ (π‘₯ = {𝑦} ∧ 𝑦 ∈ 𝐴)) β†’ (Β¬ 𝐴 ∈ Fin ∧ (π‘₯ = {𝑦} ∧ π‘₯ ∈ 𝒫 𝐴)))
14133impb 1116 . . . . . . . . . . . . . . . . . . . . . . 23 ((Β¬ 𝐴 ∈ Fin ∧ π‘₯ = {𝑦} ∧ 𝑦 ∈ 𝐴) β†’ (Β¬ 𝐴 ∈ Fin ∧ (π‘₯ = {𝑦} ∧ π‘₯ ∈ 𝒫 𝐴)))
15 3anass 1096 . . . . . . . . . . . . . . . . . . . . . . 23 ((Β¬ 𝐴 ∈ Fin ∧ π‘₯ = {𝑦} ∧ π‘₯ ∈ 𝒫 𝐴) ↔ (Β¬ 𝐴 ∈ Fin ∧ (π‘₯ = {𝑦} ∧ π‘₯ ∈ 𝒫 𝐴)))
1614, 15sylibr 233 . . . . . . . . . . . . . . . . . . . . . 22 ((Β¬ 𝐴 ∈ Fin ∧ π‘₯ = {𝑦} ∧ 𝑦 ∈ 𝐴) β†’ (Β¬ 𝐴 ∈ Fin ∧ π‘₯ = {𝑦} ∧ π‘₯ ∈ 𝒫 𝐴))
17 snfi 8995 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 {𝑦} ∈ Fin
18 eleq1 2826 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (π‘₯ = {𝑦} β†’ (π‘₯ ∈ Fin ↔ {𝑦} ∈ Fin))
1917, 18mpbiri 258 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (π‘₯ = {𝑦} β†’ π‘₯ ∈ Fin)
20 difinf 9267 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((Β¬ 𝐴 ∈ Fin ∧ π‘₯ ∈ Fin) β†’ Β¬ (𝐴 βˆ– π‘₯) ∈ Fin)
2119, 20sylan2 594 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((Β¬ 𝐴 ∈ Fin ∧ π‘₯ = {𝑦}) β†’ Β¬ (𝐴 βˆ– π‘₯) ∈ Fin)
2221orcd 872 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((Β¬ 𝐴 ∈ Fin ∧ π‘₯ = {𝑦}) β†’ (Β¬ (𝐴 βˆ– π‘₯) ∈ Fin ∨ (π‘₯ = βˆ… ∨ π‘₯ = 𝐴)))
2322anim2i 618 . . . . . . . . . . . . . . . . . . . . . . . 24 ((π‘₯ ∈ 𝒫 𝐴 ∧ (Β¬ 𝐴 ∈ Fin ∧ π‘₯ = {𝑦})) β†’ (π‘₯ ∈ 𝒫 𝐴 ∧ (Β¬ (𝐴 βˆ– π‘₯) ∈ Fin ∨ (π‘₯ = βˆ… ∨ π‘₯ = 𝐴))))
2423ancoms 460 . . . . . . . . . . . . . . . . . . . . . . 23 (((Β¬ 𝐴 ∈ Fin ∧ π‘₯ = {𝑦}) ∧ π‘₯ ∈ 𝒫 𝐴) β†’ (π‘₯ ∈ 𝒫 𝐴 ∧ (Β¬ (𝐴 βˆ– π‘₯) ∈ Fin ∨ (π‘₯ = βˆ… ∨ π‘₯ = 𝐴))))
25243impa 1111 . . . . . . . . . . . . . . . . . . . . . 22 ((Β¬ 𝐴 ∈ Fin ∧ π‘₯ = {𝑦} ∧ π‘₯ ∈ 𝒫 𝐴) β†’ (π‘₯ ∈ 𝒫 𝐴 ∧ (Β¬ (𝐴 βˆ– π‘₯) ∈ Fin ∨ (π‘₯ = βˆ… ∨ π‘₯ = 𝐴))))
2616, 25syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((Β¬ 𝐴 ∈ Fin ∧ π‘₯ = {𝑦} ∧ 𝑦 ∈ 𝐴) β†’ (π‘₯ ∈ 𝒫 𝐴 ∧ (Β¬ (𝐴 βˆ– π‘₯) ∈ Fin ∨ (π‘₯ = βˆ… ∨ π‘₯ = 𝐴))))
27 topdifinf.t . . . . . . . . . . . . . . . . . . . . . 22 𝑇 = {π‘₯ ∈ 𝒫 𝐴 ∣ (Β¬ (𝐴 βˆ– π‘₯) ∈ Fin ∨ (π‘₯ = βˆ… ∨ π‘₯ = 𝐴))}
2827reqabi 3432 . . . . . . . . . . . . . . . . . . . . 21 (π‘₯ ∈ 𝑇 ↔ (π‘₯ ∈ 𝒫 𝐴 ∧ (Β¬ (𝐴 βˆ– π‘₯) ∈ Fin ∨ (π‘₯ = βˆ… ∨ π‘₯ = 𝐴))))
2926, 28sylibr 233 . . . . . . . . . . . . . . . . . . . 20 ((Β¬ 𝐴 ∈ Fin ∧ π‘₯ = {𝑦} ∧ 𝑦 ∈ 𝐴) β†’ π‘₯ ∈ 𝑇)
30 eleq1 2826 . . . . . . . . . . . . . . . . . . . . 21 (π‘₯ = {𝑦} β†’ (π‘₯ ∈ 𝑇 ↔ {𝑦} ∈ 𝑇))
31303ad2ant2 1135 . . . . . . . . . . . . . . . . . . . 20 ((Β¬ 𝐴 ∈ Fin ∧ π‘₯ = {𝑦} ∧ 𝑦 ∈ 𝐴) β†’ (π‘₯ ∈ 𝑇 ↔ {𝑦} ∈ 𝑇))
3229, 31mpbid 231 . . . . . . . . . . . . . . . . . . 19 ((Β¬ 𝐴 ∈ Fin ∧ π‘₯ = {𝑦} ∧ 𝑦 ∈ 𝐴) β†’ {𝑦} ∈ 𝑇)
3332sbcth 3759 . . . . . . . . . . . . . . . . . 18 ({𝑦} ∈ V β†’ [{𝑦} / π‘₯]((Β¬ 𝐴 ∈ Fin ∧ π‘₯ = {𝑦} ∧ 𝑦 ∈ 𝐴) β†’ {𝑦} ∈ 𝑇))
348, 33ax-mp 5 . . . . . . . . . . . . . . . . 17 [{𝑦} / π‘₯]((Β¬ 𝐴 ∈ Fin ∧ π‘₯ = {𝑦} ∧ 𝑦 ∈ 𝐴) β†’ {𝑦} ∈ 𝑇)
35 sbcimg 3795 . . . . . . . . . . . . . . . . . 18 ({𝑦} ∈ V β†’ ([{𝑦} / π‘₯]((Β¬ 𝐴 ∈ Fin ∧ π‘₯ = {𝑦} ∧ 𝑦 ∈ 𝐴) β†’ {𝑦} ∈ 𝑇) ↔ ([{𝑦} / π‘₯](Β¬ 𝐴 ∈ Fin ∧ π‘₯ = {𝑦} ∧ 𝑦 ∈ 𝐴) β†’ [{𝑦} / π‘₯]{𝑦} ∈ 𝑇)))
368, 35ax-mp 5 . . . . . . . . . . . . . . . . 17 ([{𝑦} / π‘₯]((Β¬ 𝐴 ∈ Fin ∧ π‘₯ = {𝑦} ∧ 𝑦 ∈ 𝐴) β†’ {𝑦} ∈ 𝑇) ↔ ([{𝑦} / π‘₯](Β¬ 𝐴 ∈ Fin ∧ π‘₯ = {𝑦} ∧ 𝑦 ∈ 𝐴) β†’ [{𝑦} / π‘₯]{𝑦} ∈ 𝑇))
3734, 36mpbi 229 . . . . . . . . . . . . . . . 16 ([{𝑦} / π‘₯](Β¬ 𝐴 ∈ Fin ∧ π‘₯ = {𝑦} ∧ 𝑦 ∈ 𝐴) β†’ [{𝑦} / π‘₯]{𝑦} ∈ 𝑇)
38 sbc3an 3814 . . . . . . . . . . . . . . . . . 18 ([{𝑦} / π‘₯](Β¬ 𝐴 ∈ Fin ∧ π‘₯ = {𝑦} ∧ 𝑦 ∈ 𝐴) ↔ ([{𝑦} / π‘₯] Β¬ 𝐴 ∈ Fin ∧ [{𝑦} / π‘₯]π‘₯ = {𝑦} ∧ [{𝑦} / π‘₯]𝑦 ∈ 𝐴))
39 sbcg 3823 . . . . . . . . . . . . . . . . . . . 20 ({𝑦} ∈ V β†’ ([{𝑦} / π‘₯] Β¬ 𝐴 ∈ Fin ↔ Β¬ 𝐴 ∈ Fin))
408, 39ax-mp 5 . . . . . . . . . . . . . . . . . . 19 ([{𝑦} / π‘₯] Β¬ 𝐴 ∈ Fin ↔ Β¬ 𝐴 ∈ Fin)
41403anbi1i 1158 . . . . . . . . . . . . . . . . . 18 (([{𝑦} / π‘₯] Β¬ 𝐴 ∈ Fin ∧ [{𝑦} / π‘₯]π‘₯ = {𝑦} ∧ [{𝑦} / π‘₯]𝑦 ∈ 𝐴) ↔ (Β¬ 𝐴 ∈ Fin ∧ [{𝑦} / π‘₯]π‘₯ = {𝑦} ∧ [{𝑦} / π‘₯]𝑦 ∈ 𝐴))
42 eqsbc1 3793 . . . . . . . . . . . . . . . . . . . 20 ({𝑦} ∈ V β†’ ([{𝑦} / π‘₯]π‘₯ = {𝑦} ↔ {𝑦} = {𝑦}))
438, 42ax-mp 5 . . . . . . . . . . . . . . . . . . 19 ([{𝑦} / π‘₯]π‘₯ = {𝑦} ↔ {𝑦} = {𝑦})
44433anbi2i 1159 . . . . . . . . . . . . . . . . . 18 ((Β¬ 𝐴 ∈ Fin ∧ [{𝑦} / π‘₯]π‘₯ = {𝑦} ∧ [{𝑦} / π‘₯]𝑦 ∈ 𝐴) ↔ (Β¬ 𝐴 ∈ Fin ∧ {𝑦} = {𝑦} ∧ [{𝑦} / π‘₯]𝑦 ∈ 𝐴))
4538, 41, 443bitri 297 . . . . . . . . . . . . . . . . 17 ([{𝑦} / π‘₯](Β¬ 𝐴 ∈ Fin ∧ π‘₯ = {𝑦} ∧ 𝑦 ∈ 𝐴) ↔ (Β¬ 𝐴 ∈ Fin ∧ {𝑦} = {𝑦} ∧ [{𝑦} / π‘₯]𝑦 ∈ 𝐴))
46 sbcg 3823 . . . . . . . . . . . . . . . . . . 19 ({𝑦} ∈ V β†’ ([{𝑦} / π‘₯]𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
478, 46ax-mp 5 . . . . . . . . . . . . . . . . . 18 ([{𝑦} / π‘₯]𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
48473anbi3i 1160 . . . . . . . . . . . . . . . . 17 ((Β¬ 𝐴 ∈ Fin ∧ {𝑦} = {𝑦} ∧ [{𝑦} / π‘₯]𝑦 ∈ 𝐴) ↔ (Β¬ 𝐴 ∈ Fin ∧ {𝑦} = {𝑦} ∧ 𝑦 ∈ 𝐴))
4945, 48bitri 275 . . . . . . . . . . . . . . . 16 ([{𝑦} / π‘₯](Β¬ 𝐴 ∈ Fin ∧ π‘₯ = {𝑦} ∧ 𝑦 ∈ 𝐴) ↔ (Β¬ 𝐴 ∈ Fin ∧ {𝑦} = {𝑦} ∧ 𝑦 ∈ 𝐴))
50 sbcg 3823 . . . . . . . . . . . . . . . . 17 ({𝑦} ∈ V β†’ ([{𝑦} / π‘₯]{𝑦} ∈ 𝑇 ↔ {𝑦} ∈ 𝑇))
518, 50ax-mp 5 . . . . . . . . . . . . . . . 16 ([{𝑦} / π‘₯]{𝑦} ∈ 𝑇 ↔ {𝑦} ∈ 𝑇)
5237, 49, 513imtr3i 291 . . . . . . . . . . . . . . 15 ((Β¬ 𝐴 ∈ Fin ∧ {𝑦} = {𝑦} ∧ 𝑦 ∈ 𝐴) β†’ {𝑦} ∈ 𝑇)
537, 52mp3an2 1450 . . . . . . . . . . . . . 14 ((Β¬ 𝐴 ∈ Fin ∧ 𝑦 ∈ 𝐴) β†’ {𝑦} ∈ 𝑇)
5453ex 414 . . . . . . . . . . . . 13 (Β¬ 𝐴 ∈ Fin β†’ (𝑦 ∈ 𝐴 β†’ {𝑦} ∈ 𝑇))
5554pm4.71d 563 . . . . . . . . . . . 12 (Β¬ 𝐴 ∈ Fin β†’ (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ 𝐴 ∧ {𝑦} ∈ 𝑇)))
5655anbi1d 631 . . . . . . . . . . 11 (Β¬ 𝐴 ∈ Fin β†’ ((𝑦 ∈ 𝐴 ∧ 𝑒 = {𝑦}) ↔ ((𝑦 ∈ 𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑒 = {𝑦})))
5756exbidv 1925 . . . . . . . . . 10 (Β¬ 𝐴 ∈ Fin β†’ (βˆƒπ‘¦(𝑦 ∈ 𝐴 ∧ 𝑒 = {𝑦}) ↔ βˆƒπ‘¦((𝑦 ∈ 𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑒 = {𝑦})))
586, 57bitrid 283 . . . . . . . . 9 (Β¬ 𝐴 ∈ Fin β†’ (𝑒 ∈ {𝑒 ∣ βˆƒπ‘¦ ∈ 𝐴 𝑒 = {𝑦}} ↔ βˆƒπ‘¦((𝑦 ∈ 𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑒 = {𝑦})))
59 anass 470 . . . . . . . . . . 11 (((𝑦 ∈ 𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑒 = {𝑦}) ↔ (𝑦 ∈ 𝐴 ∧ ({𝑦} ∈ 𝑇 ∧ 𝑒 = {𝑦})))
6059exbii 1851 . . . . . . . . . 10 (βˆƒπ‘¦((𝑦 ∈ 𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑒 = {𝑦}) ↔ βˆƒπ‘¦(𝑦 ∈ 𝐴 ∧ ({𝑦} ∈ 𝑇 ∧ 𝑒 = {𝑦})))
61 exsimpr 1873 . . . . . . . . . 10 (βˆƒπ‘¦(𝑦 ∈ 𝐴 ∧ ({𝑦} ∈ 𝑇 ∧ 𝑒 = {𝑦})) β†’ βˆƒπ‘¦({𝑦} ∈ 𝑇 ∧ 𝑒 = {𝑦}))
6260, 61sylbi 216 . . . . . . . . 9 (βˆƒπ‘¦((𝑦 ∈ 𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑒 = {𝑦}) β†’ βˆƒπ‘¦({𝑦} ∈ 𝑇 ∧ 𝑒 = {𝑦}))
6358, 62syl6bi 253 . . . . . . . 8 (Β¬ 𝐴 ∈ Fin β†’ (𝑒 ∈ {𝑒 ∣ βˆƒπ‘¦ ∈ 𝐴 𝑒 = {𝑦}} β†’ βˆƒπ‘¦({𝑦} ∈ 𝑇 ∧ 𝑒 = {𝑦})))
64 ancom 462 . . . . . . . . . 10 (({𝑦} ∈ 𝑇 ∧ 𝑒 = {𝑦}) ↔ (𝑒 = {𝑦} ∧ {𝑦} ∈ 𝑇))
65 eleq1 2826 . . . . . . . . . . 11 (𝑒 = {𝑦} β†’ (𝑒 ∈ 𝑇 ↔ {𝑦} ∈ 𝑇))
6665pm5.32i 576 . . . . . . . . . 10 ((𝑒 = {𝑦} ∧ 𝑒 ∈ 𝑇) ↔ (𝑒 = {𝑦} ∧ {𝑦} ∈ 𝑇))
6764, 66bitr4i 278 . . . . . . . . 9 (({𝑦} ∈ 𝑇 ∧ 𝑒 = {𝑦}) ↔ (𝑒 = {𝑦} ∧ 𝑒 ∈ 𝑇))
6867exbii 1851 . . . . . . . 8 (βˆƒπ‘¦({𝑦} ∈ 𝑇 ∧ 𝑒 = {𝑦}) ↔ βˆƒπ‘¦(𝑒 = {𝑦} ∧ 𝑒 ∈ 𝑇))
6963, 68syl6ib 251 . . . . . . 7 (Β¬ 𝐴 ∈ Fin β†’ (𝑒 ∈ {𝑒 ∣ βˆƒπ‘¦ ∈ 𝐴 𝑒 = {𝑦}} β†’ βˆƒπ‘¦(𝑒 = {𝑦} ∧ 𝑒 ∈ 𝑇)))
70 exsimpr 1873 . . . . . . 7 (βˆƒπ‘¦(𝑒 = {𝑦} ∧ 𝑒 ∈ 𝑇) β†’ βˆƒπ‘¦ 𝑒 ∈ 𝑇)
7169, 70syl6 35 . . . . . 6 (Β¬ 𝐴 ∈ Fin β†’ (𝑒 ∈ {𝑒 ∣ βˆƒπ‘¦ ∈ 𝐴 𝑒 = {𝑦}} β†’ βˆƒπ‘¦ 𝑒 ∈ 𝑇))
72 ax5e 1916 . . . . . 6 (βˆƒπ‘¦ 𝑒 ∈ 𝑇 β†’ 𝑒 ∈ 𝑇)
7371, 72syl6 35 . . . . 5 (Β¬ 𝐴 ∈ Fin β†’ (𝑒 ∈ {𝑒 ∣ βˆƒπ‘¦ ∈ 𝐴 𝑒 = {𝑦}} β†’ 𝑒 ∈ 𝑇))
741, 2, 3, 73ssrd 3954 . . . 4 (Β¬ 𝐴 ∈ Fin β†’ {𝑒 ∣ βˆƒπ‘¦ ∈ 𝐴 𝑒 = {𝑦}} βŠ† 𝑇)
75 eqid 2737 . . . . 5 {𝑒 ∣ βˆƒπ‘¦ ∈ 𝐴 𝑒 = {𝑦}} = {𝑒 ∣ βˆƒπ‘¦ ∈ 𝐴 𝑒 = {𝑦}}
7675dissneq 35841 . . . 4 (({𝑒 ∣ βˆƒπ‘¦ ∈ 𝐴 𝑒 = {𝑦}} βŠ† 𝑇 ∧ 𝑇 ∈ (TopOnβ€˜π΄)) β†’ 𝑇 = 𝒫 𝐴)
7774, 76sylan 581 . . 3 ((Β¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOnβ€˜π΄)) β†’ 𝑇 = 𝒫 𝐴)
78 nfielex 9224 . . . . 5 (Β¬ 𝐴 ∈ Fin β†’ βˆƒπ‘¦ 𝑦 ∈ 𝐴)
7978adantr 482 . . . 4 ((Β¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOnβ€˜π΄)) β†’ βˆƒπ‘¦ 𝑦 ∈ 𝐴)
80 difss 4096 . . . . . . 7 (𝐴 βˆ– {𝑦}) βŠ† 𝐴
81 elfvex 6885 . . . . . . . 8 (𝑇 ∈ (TopOnβ€˜π΄) β†’ 𝐴 ∈ V)
82 difexg 5289 . . . . . . . 8 (𝐴 ∈ V β†’ (𝐴 βˆ– {𝑦}) ∈ V)
83 elpwg 4568 . . . . . . . 8 ((𝐴 βˆ– {𝑦}) ∈ V β†’ ((𝐴 βˆ– {𝑦}) ∈ 𝒫 𝐴 ↔ (𝐴 βˆ– {𝑦}) βŠ† 𝐴))
8481, 82, 833syl 18 . . . . . . 7 (𝑇 ∈ (TopOnβ€˜π΄) β†’ ((𝐴 βˆ– {𝑦}) ∈ 𝒫 𝐴 ↔ (𝐴 βˆ– {𝑦}) βŠ† 𝐴))
8580, 84mpbiri 258 . . . . . 6 (𝑇 ∈ (TopOnβ€˜π΄) β†’ (𝐴 βˆ– {𝑦}) ∈ 𝒫 𝐴)
8685adantl 483 . . . . 5 ((Β¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOnβ€˜π΄)) β†’ (𝐴 βˆ– {𝑦}) ∈ 𝒫 𝐴)
87 difinf 9267 . . . . . . . . . . . 12 ((Β¬ 𝐴 ∈ Fin ∧ {𝑦} ∈ Fin) β†’ Β¬ (𝐴 βˆ– {𝑦}) ∈ Fin)
8817, 87mpan2 690 . . . . . . . . . . 11 (Β¬ 𝐴 ∈ Fin β†’ Β¬ (𝐴 βˆ– {𝑦}) ∈ Fin)
89 0fin 9122 . . . . . . . . . . . 12 βˆ… ∈ Fin
90 eleq1 2826 . . . . . . . . . . . 12 ((𝐴 βˆ– {𝑦}) = βˆ… β†’ ((𝐴 βˆ– {𝑦}) ∈ Fin ↔ βˆ… ∈ Fin))
9189, 90mpbiri 258 . . . . . . . . . . 11 ((𝐴 βˆ– {𝑦}) = βˆ… β†’ (𝐴 βˆ– {𝑦}) ∈ Fin)
9288, 91nsyl 140 . . . . . . . . . 10 (Β¬ 𝐴 ∈ Fin β†’ Β¬ (𝐴 βˆ– {𝑦}) = βˆ…)
9392ad2antrl 727 . . . . . . . . 9 ((𝑦 ∈ 𝐴 ∧ (Β¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOnβ€˜π΄))) β†’ Β¬ (𝐴 βˆ– {𝑦}) = βˆ…)
94 vsnid 4628 . . . . . . . . . . . . . 14 𝑦 ∈ {𝑦}
95 inelcm 4429 . . . . . . . . . . . . . 14 ((𝑦 ∈ 𝐴 ∧ 𝑦 ∈ {𝑦}) β†’ (𝐴 ∩ {𝑦}) β‰  βˆ…)
9694, 95mpan2 690 . . . . . . . . . . . . 13 (𝑦 ∈ 𝐴 β†’ (𝐴 ∩ {𝑦}) β‰  βˆ…)
97 disj4 4423 . . . . . . . . . . . . . 14 ((𝐴 ∩ {𝑦}) = βˆ… ↔ Β¬ (𝐴 βˆ– {𝑦}) ⊊ 𝐴)
9897necon2abii 2995 . . . . . . . . . . . . 13 ((𝐴 βˆ– {𝑦}) ⊊ 𝐴 ↔ (𝐴 ∩ {𝑦}) β‰  βˆ…)
9996, 98sylibr 233 . . . . . . . . . . . 12 (𝑦 ∈ 𝐴 β†’ (𝐴 βˆ– {𝑦}) ⊊ 𝐴)
10099pssned 4063 . . . . . . . . . . 11 (𝑦 ∈ 𝐴 β†’ (𝐴 βˆ– {𝑦}) β‰  𝐴)
101100adantr 482 . . . . . . . . . 10 ((𝑦 ∈ 𝐴 ∧ (Β¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOnβ€˜π΄))) β†’ (𝐴 βˆ– {𝑦}) β‰  𝐴)
102101neneqd 2949 . . . . . . . . 9 ((𝑦 ∈ 𝐴 ∧ (Β¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOnβ€˜π΄))) β†’ Β¬ (𝐴 βˆ– {𝑦}) = 𝐴)
10393, 102jca 513 . . . . . . . 8 ((𝑦 ∈ 𝐴 ∧ (Β¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOnβ€˜π΄))) β†’ (Β¬ (𝐴 βˆ– {𝑦}) = βˆ… ∧ Β¬ (𝐴 βˆ– {𝑦}) = 𝐴))
104 pm4.56 988 . . . . . . . 8 ((Β¬ (𝐴 βˆ– {𝑦}) = βˆ… ∧ Β¬ (𝐴 βˆ– {𝑦}) = 𝐴) ↔ Β¬ ((𝐴 βˆ– {𝑦}) = βˆ… ∨ (𝐴 βˆ– {𝑦}) = 𝐴))
105103, 104sylib 217 . . . . . . 7 ((𝑦 ∈ 𝐴 ∧ (Β¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOnβ€˜π΄))) β†’ Β¬ ((𝐴 βˆ– {𝑦}) = βˆ… ∨ (𝐴 βˆ– {𝑦}) = 𝐴))
106 difeq2 4081 . . . . . . . . . . . . . 14 (π‘₯ = (𝐴 βˆ– {𝑦}) β†’ (𝐴 βˆ– π‘₯) = (𝐴 βˆ– (𝐴 βˆ– {𝑦})))
107106eleq1d 2823 . . . . . . . . . . . . 13 (π‘₯ = (𝐴 βˆ– {𝑦}) β†’ ((𝐴 βˆ– π‘₯) ∈ Fin ↔ (𝐴 βˆ– (𝐴 βˆ– {𝑦})) ∈ Fin))
108107notbid 318 . . . . . . . . . . . 12 (π‘₯ = (𝐴 βˆ– {𝑦}) β†’ (Β¬ (𝐴 βˆ– π‘₯) ∈ Fin ↔ Β¬ (𝐴 βˆ– (𝐴 βˆ– {𝑦})) ∈ Fin))
109 eqeq1 2741 . . . . . . . . . . . . 13 (π‘₯ = (𝐴 βˆ– {𝑦}) β†’ (π‘₯ = βˆ… ↔ (𝐴 βˆ– {𝑦}) = βˆ…))
110 eqeq1 2741 . . . . . . . . . . . . 13 (π‘₯ = (𝐴 βˆ– {𝑦}) β†’ (π‘₯ = 𝐴 ↔ (𝐴 βˆ– {𝑦}) = 𝐴))
111109, 110orbi12d 918 . . . . . . . . . . . 12 (π‘₯ = (𝐴 βˆ– {𝑦}) β†’ ((π‘₯ = βˆ… ∨ π‘₯ = 𝐴) ↔ ((𝐴 βˆ– {𝑦}) = βˆ… ∨ (𝐴 βˆ– {𝑦}) = 𝐴)))
112108, 111orbi12d 918 . . . . . . . . . . 11 (π‘₯ = (𝐴 βˆ– {𝑦}) β†’ ((Β¬ (𝐴 βˆ– π‘₯) ∈ Fin ∨ (π‘₯ = βˆ… ∨ π‘₯ = 𝐴)) ↔ (Β¬ (𝐴 βˆ– (𝐴 βˆ– {𝑦})) ∈ Fin ∨ ((𝐴 βˆ– {𝑦}) = βˆ… ∨ (𝐴 βˆ– {𝑦}) = 𝐴))))
113112, 27elrab2 3653 . . . . . . . . . 10 ((𝐴 βˆ– {𝑦}) ∈ 𝑇 ↔ ((𝐴 βˆ– {𝑦}) ∈ 𝒫 𝐴 ∧ (Β¬ (𝐴 βˆ– (𝐴 βˆ– {𝑦})) ∈ Fin ∨ ((𝐴 βˆ– {𝑦}) = βˆ… ∨ (𝐴 βˆ– {𝑦}) = 𝐴))))
11485biantrurd 534 . . . . . . . . . 10 (𝑇 ∈ (TopOnβ€˜π΄) β†’ ((Β¬ (𝐴 βˆ– (𝐴 βˆ– {𝑦})) ∈ Fin ∨ ((𝐴 βˆ– {𝑦}) = βˆ… ∨ (𝐴 βˆ– {𝑦}) = 𝐴)) ↔ ((𝐴 βˆ– {𝑦}) ∈ 𝒫 𝐴 ∧ (Β¬ (𝐴 βˆ– (𝐴 βˆ– {𝑦})) ∈ Fin ∨ ((𝐴 βˆ– {𝑦}) = βˆ… ∨ (𝐴 βˆ– {𝑦}) = 𝐴)))))
115113, 114bitr4id 290 . . . . . . . . 9 (𝑇 ∈ (TopOnβ€˜π΄) β†’ ((𝐴 βˆ– {𝑦}) ∈ 𝑇 ↔ (Β¬ (𝐴 βˆ– (𝐴 βˆ– {𝑦})) ∈ Fin ∨ ((𝐴 βˆ– {𝑦}) = βˆ… ∨ (𝐴 βˆ– {𝑦}) = 𝐴))))
116 dfin4 4232 . . . . . . . . . . 11 (𝐴 ∩ {𝑦}) = (𝐴 βˆ– (𝐴 βˆ– {𝑦}))
117 inss2 4194 . . . . . . . . . . . 12 (𝐴 ∩ {𝑦}) βŠ† {𝑦}
118 ssfi 9124 . . . . . . . . . . . 12 (({𝑦} ∈ Fin ∧ (𝐴 ∩ {𝑦}) βŠ† {𝑦}) β†’ (𝐴 ∩ {𝑦}) ∈ Fin)
11917, 117, 118mp2an 691 . . . . . . . . . . 11 (𝐴 ∩ {𝑦}) ∈ Fin
120116, 119eqeltrri 2835 . . . . . . . . . 10 (𝐴 βˆ– (𝐴 βˆ– {𝑦})) ∈ Fin
121 biortn 937 . . . . . . . . . 10 ((𝐴 βˆ– (𝐴 βˆ– {𝑦})) ∈ Fin β†’ (((𝐴 βˆ– {𝑦}) = βˆ… ∨ (𝐴 βˆ– {𝑦}) = 𝐴) ↔ (Β¬ (𝐴 βˆ– (𝐴 βˆ– {𝑦})) ∈ Fin ∨ ((𝐴 βˆ– {𝑦}) = βˆ… ∨ (𝐴 βˆ– {𝑦}) = 𝐴))))
122120, 121ax-mp 5 . . . . . . . . 9 (((𝐴 βˆ– {𝑦}) = βˆ… ∨ (𝐴 βˆ– {𝑦}) = 𝐴) ↔ (Β¬ (𝐴 βˆ– (𝐴 βˆ– {𝑦})) ∈ Fin ∨ ((𝐴 βˆ– {𝑦}) = βˆ… ∨ (𝐴 βˆ– {𝑦}) = 𝐴)))
123115, 122bitr4di 289 . . . . . . . 8 (𝑇 ∈ (TopOnβ€˜π΄) β†’ ((𝐴 βˆ– {𝑦}) ∈ 𝑇 ↔ ((𝐴 βˆ– {𝑦}) = βˆ… ∨ (𝐴 βˆ– {𝑦}) = 𝐴)))
124123ad2antll 728 . . . . . . 7 ((𝑦 ∈ 𝐴 ∧ (Β¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOnβ€˜π΄))) β†’ ((𝐴 βˆ– {𝑦}) ∈ 𝑇 ↔ ((𝐴 βˆ– {𝑦}) = βˆ… ∨ (𝐴 βˆ– {𝑦}) = 𝐴)))
125105, 124mtbird 325 . . . . . 6 ((𝑦 ∈ 𝐴 ∧ (Β¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOnβ€˜π΄))) β†’ Β¬ (𝐴 βˆ– {𝑦}) ∈ 𝑇)
126125expcom 415 . . . . 5 ((Β¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOnβ€˜π΄)) β†’ (𝑦 ∈ 𝐴 β†’ Β¬ (𝐴 βˆ– {𝑦}) ∈ 𝑇))
127 nelneq2 2863 . . . . . 6 (((𝐴 βˆ– {𝑦}) ∈ 𝒫 𝐴 ∧ Β¬ (𝐴 βˆ– {𝑦}) ∈ 𝑇) β†’ Β¬ 𝒫 𝐴 = 𝑇)
128 eqcom 2744 . . . . . 6 (𝑇 = 𝒫 𝐴 ↔ 𝒫 𝐴 = 𝑇)
129127, 128sylnibr 329 . . . . 5 (((𝐴 βˆ– {𝑦}) ∈ 𝒫 𝐴 ∧ Β¬ (𝐴 βˆ– {𝑦}) ∈ 𝑇) β†’ Β¬ 𝑇 = 𝒫 𝐴)
13086, 126, 129syl6an 683 . . . 4 ((Β¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOnβ€˜π΄)) β†’ (𝑦 ∈ 𝐴 β†’ Β¬ 𝑇 = 𝒫 𝐴))
13179, 130exellimddv 35845 . . 3 ((Β¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOnβ€˜π΄)) β†’ Β¬ 𝑇 = 𝒫 𝐴)
13277, 131pm2.65da 816 . 2 (Β¬ 𝐴 ∈ Fin β†’ Β¬ 𝑇 ∈ (TopOnβ€˜π΄))
133132con4i 114 1 (𝑇 ∈ (TopOnβ€˜π΄) β†’ 𝐴 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {cab 2714   β‰  wne 2944  βˆƒwrex 3074  {crab 3410  Vcvv 3448  [wsbc 3744   βˆ– cdif 3912   ∩ cin 3914   βŠ† wss 3915   ⊊ wpss 3916  βˆ…c0 4287  π’« cpw 4565  {csn 4591  β€˜cfv 6501  Fincfn 8890  TopOnctopon 22275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-om 7808  df-1o 8417  df-en 8891  df-fin 8894  df-topgen 17332  df-top 22259  df-topon 22276
This theorem is referenced by:  topdifinffin  35848
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