Theorem topdifinffinlem 37554

Description: This is the core of the proof of topdifinffin 37555, 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 ∣ (¬ (𝐴 ∖ 𝑥) ∈ 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.

Step	Hyp	Ref	Expression
1		nfv 1916	. . . . 5 ⊢ Ⅎ𝑢 ¬ 𝐴 ∈ Fin
2		nfab1 2901	. . . . 5 ⊢ Ⅎ𝑢{𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}}
3		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑢𝑇
4		abid 2719	. . . . . . . . . . 11 ⊢ (𝑢 ∈ {𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}} ↔ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦})
5		df-rex 3062	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝐴 𝑢 = {𝑦} ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑢 = {𝑦}))
6	4, 5	bitri 275	. . . . . . . . . 10 ⊢ (𝑢 ∈ {𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}} ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑢 = {𝑦}))
7		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ {𝑦} = {𝑦}
8		vsnex 5380	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑦} ∈ V
9		snelpwi 5393	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ∈ 𝒫 𝐴)
10		eleq1 2825	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 = {𝑦} → (𝑥 ∈ 𝒫 𝐴 ↔ {𝑦} ∈ 𝒫 𝐴))
11	9, 10	imbitrrid 246	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = {𝑦} → (𝑦 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴))
12	11	imdistani 568	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴) → (𝑥 = {𝑦} ∧ 𝑥 ∈ 𝒫 𝐴))
13	12	anim2i 618	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴)) → (¬ 𝐴 ∈ Fin ∧ (𝑥 = {𝑦} ∧ 𝑥 ∈ 𝒫 𝐴)))
14	13	3impb 1115	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴) → (¬ 𝐴 ∈ Fin ∧ (𝑥 = {𝑦} ∧ 𝑥 ∈ 𝒫 𝐴)))
15		3anass 1095	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑥 ∈ 𝒫 𝐴) ↔ (¬ 𝐴 ∈ Fin ∧ (𝑥 = {𝑦} ∧ 𝑥 ∈ 𝒫 𝐴)))
16	14, 15	sylibr 234	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴) → (¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑥 ∈ 𝒫 𝐴))
17		snfi 8984	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ {𝑦} ∈ Fin
18		eleq1 2825	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 = {𝑦} → (𝑥 ∈ Fin ↔ {𝑦} ∈ Fin))
19	17, 18	mpbiri 258	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 = {𝑦} → 𝑥 ∈ Fin)
20		difinf 9215	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑥 ∈ Fin) → ¬ (𝐴 ∖ 𝑥) ∈ Fin)
21	19, 20	sylan2 594	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦}) → ¬ (𝐴 ∖ 𝑥) ∈ Fin)
22	21	orcd 874	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦}) → (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)))
23	22	anim2i 618	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦})) → (𝑥 ∈ 𝒫 𝐴 ∧ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
24	23	ancoms 458	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦}) ∧ 𝑥 ∈ 𝒫 𝐴) → (𝑥 ∈ 𝒫 𝐴 ∧ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
25	24	3impa 1110	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑥 ∈ 𝒫 𝐴) → (𝑥 ∈ 𝒫 𝐴 ∧ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
26	16, 25	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ 𝒫 𝐴 ∧ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
27		topdifinf.t	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}
28	27	reqabi 3423	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ 𝑇 ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
29	26, 28	sylibr 234	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝑇)
30		eleq1 2825	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = {𝑦} → (𝑥 ∈ 𝑇 ↔ {𝑦} ∈ 𝑇))
31	30	3ad2ant2 1135	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ 𝑇 ↔ {𝑦} ∈ 𝑇))
32	29, 31	mpbid 232	. . . . . . . . . . . . . . . . . . 19 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴) → {𝑦} ∈ 𝑇)
33	32	sbcth 3756	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑦} ∈ V → [{𝑦} / 𝑥]((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴) → {𝑦} ∈ 𝑇))
34	8, 33	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ [{𝑦} / 𝑥]((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴) → {𝑦} ∈ 𝑇)
35		sbcimg 3790	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑦} ∈ V → ([{𝑦} / 𝑥]((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴) → {𝑦} ∈ 𝑇) ↔ ([{𝑦} / 𝑥](¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴) → [{𝑦} / 𝑥]{𝑦} ∈ 𝑇)))
36	8, 35	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ([{𝑦} / 𝑥]((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴) → {𝑦} ∈ 𝑇) ↔ ([{𝑦} / 𝑥](¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴) → [{𝑦} / 𝑥]{𝑦} ∈ 𝑇))
37	34, 36	mpbi 230	. . . . . . . . . . . . . . . 16 ⊢ ([{𝑦} / 𝑥](¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴) → [{𝑦} / 𝑥]{𝑦} ∈ 𝑇)
38		sbc3an 3806	. . . . . . . . . . . . . . . . . 18 ⊢ ([{𝑦} / 𝑥](¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴) ↔ ([{𝑦} / 𝑥] ¬ 𝐴 ∈ Fin ∧ [{𝑦} / 𝑥]𝑥 = {𝑦} ∧ [{𝑦} / 𝑥]𝑦 ∈ 𝐴))
39		sbcg 3814	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({𝑦} ∈ V → ([{𝑦} / 𝑥] ¬ 𝐴 ∈ Fin ↔ ¬ 𝐴 ∈ Fin))
40	8, 39	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ ([{𝑦} / 𝑥] ¬ 𝐴 ∈ Fin ↔ ¬ 𝐴 ∈ Fin)
41	40	3anbi1i 1158	. . . . . . . . . . . . . . . . . 18 ⊢ (([{𝑦} / 𝑥] ¬ 𝐴 ∈ Fin ∧ [{𝑦} / 𝑥]𝑥 = {𝑦} ∧ [{𝑦} / 𝑥]𝑦 ∈ 𝐴) ↔ (¬ 𝐴 ∈ Fin ∧ [{𝑦} / 𝑥]𝑥 = {𝑦} ∧ [{𝑦} / 𝑥]𝑦 ∈ 𝐴))
42		eqsbc1 3788	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({𝑦} ∈ V → ([{𝑦} / 𝑥]𝑥 = {𝑦} ↔ {𝑦} = {𝑦}))
43	8, 42	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ ([{𝑦} / 𝑥]𝑥 = {𝑦} ↔ {𝑦} = {𝑦})
44	43	3anbi2i 1159	. . . . . . . . . . . . . . . . . 18 ⊢ ((¬ 𝐴 ∈ Fin ∧ [{𝑦} / 𝑥]𝑥 = {𝑦} ∧ [{𝑦} / 𝑥]𝑦 ∈ 𝐴) ↔ (¬ 𝐴 ∈ Fin ∧ {𝑦} = {𝑦} ∧ [{𝑦} / 𝑥]𝑦 ∈ 𝐴))
45	38, 41, 44	3bitri 297	. . . . . . . . . . . . . . . . 17 ⊢ ([{𝑦} / 𝑥](¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴) ↔ (¬ 𝐴 ∈ Fin ∧ {𝑦} = {𝑦} ∧ [{𝑦} / 𝑥]𝑦 ∈ 𝐴))
46		sbcg 3814	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑦} ∈ V → ([{𝑦} / 𝑥]𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
47	8, 46	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ ([{𝑦} / 𝑥]𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
48	47	3anbi3i 1160	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ 𝐴 ∈ Fin ∧ {𝑦} = {𝑦} ∧ [{𝑦} / 𝑥]𝑦 ∈ 𝐴) ↔ (¬ 𝐴 ∈ Fin ∧ {𝑦} = {𝑦} ∧ 𝑦 ∈ 𝐴))
49	45, 48	bitri 275	. . . . . . . . . . . . . . . 16 ⊢ ([{𝑦} / 𝑥](¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴) ↔ (¬ 𝐴 ∈ Fin ∧ {𝑦} = {𝑦} ∧ 𝑦 ∈ 𝐴))
50		sbcg 3814	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑦} ∈ V → ([{𝑦} / 𝑥]{𝑦} ∈ 𝑇 ↔ {𝑦} ∈ 𝑇))
51	8, 50	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ([{𝑦} / 𝑥]{𝑦} ∈ 𝑇 ↔ {𝑦} ∈ 𝑇)
52	37, 49, 51	3imtr3i 291	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝐴 ∈ Fin ∧ {𝑦} = {𝑦} ∧ 𝑦 ∈ 𝐴) → {𝑦} ∈ 𝑇)
53	7, 52	mp3an2 1452	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑦 ∈ 𝐴) → {𝑦} ∈ 𝑇)
54	53	ex 412	. . . . . . . . . . . . 13 ⊢ (¬ 𝐴 ∈ Fin → (𝑦 ∈ 𝐴 → {𝑦} ∈ 𝑇))
55	54	pm4.71d 561	. . . . . . . . . . . 12 ⊢ (¬ 𝐴 ∈ Fin → (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ 𝐴 ∧ {𝑦} ∈ 𝑇)))
56	55	anbi1d 632	. . . . . . . . . . 11 ⊢ (¬ 𝐴 ∈ Fin → ((𝑦 ∈ 𝐴 ∧ 𝑢 = {𝑦}) ↔ ((𝑦 ∈ 𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑢 = {𝑦})))
57	56	exbidv 1923	. . . . . . . . . 10 ⊢ (¬ 𝐴 ∈ Fin → (∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑢 = {𝑦}) ↔ ∃𝑦((𝑦 ∈ 𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑢 = {𝑦})))
58	6, 57	bitrid 283	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ Fin → (𝑢 ∈ {𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}} ↔ ∃𝑦((𝑦 ∈ 𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑢 = {𝑦})))
59		anass 468	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ 𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑢 = {𝑦}) ↔ (𝑦 ∈ 𝐴 ∧ ({𝑦} ∈ 𝑇 ∧ 𝑢 = {𝑦})))
60	59	exbii 1850	. . . . . . . . . 10 ⊢ (∃𝑦((𝑦 ∈ 𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑢 = {𝑦}) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ({𝑦} ∈ 𝑇 ∧ 𝑢 = {𝑦})))
61		exsimpr 1871	. . . . . . . . . 10 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ ({𝑦} ∈ 𝑇 ∧ 𝑢 = {𝑦})) → ∃𝑦({𝑦} ∈ 𝑇 ∧ 𝑢 = {𝑦}))
62	60, 61	sylbi 217	. . . . . . . . 9 ⊢ (∃𝑦((𝑦 ∈ 𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑢 = {𝑦}) → ∃𝑦({𝑦} ∈ 𝑇 ∧ 𝑢 = {𝑦}))
63	58, 62	biimtrdi 253	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ Fin → (𝑢 ∈ {𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}} → ∃𝑦({𝑦} ∈ 𝑇 ∧ 𝑢 = {𝑦})))
64		ancom 460	. . . . . . . . . 10 ⊢ (({𝑦} ∈ 𝑇 ∧ 𝑢 = {𝑦}) ↔ (𝑢 = {𝑦} ∧ {𝑦} ∈ 𝑇))
65		eleq1 2825	. . . . . . . . . . 11 ⊢ (𝑢 = {𝑦} → (𝑢 ∈ 𝑇 ↔ {𝑦} ∈ 𝑇))
66	65	pm5.32i 574	. . . . . . . . . 10 ⊢ ((𝑢 = {𝑦} ∧ 𝑢 ∈ 𝑇) ↔ (𝑢 = {𝑦} ∧ {𝑦} ∈ 𝑇))
67	64, 66	bitr4i 278	. . . . . . . . 9 ⊢ (({𝑦} ∈ 𝑇 ∧ 𝑢 = {𝑦}) ↔ (𝑢 = {𝑦} ∧ 𝑢 ∈ 𝑇))
68	67	exbii 1850	. . . . . . . 8 ⊢ (∃𝑦({𝑦} ∈ 𝑇 ∧ 𝑢 = {𝑦}) ↔ ∃𝑦(𝑢 = {𝑦} ∧ 𝑢 ∈ 𝑇))
69	63, 68	imbitrdi 251	. . . . . . 7 ⊢ (¬ 𝐴 ∈ Fin → (𝑢 ∈ {𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}} → ∃𝑦(𝑢 = {𝑦} ∧ 𝑢 ∈ 𝑇)))
70		exsimpr 1871	. . . . . . 7 ⊢ (∃𝑦(𝑢 = {𝑦} ∧ 𝑢 ∈ 𝑇) → ∃𝑦 𝑢 ∈ 𝑇)
71	69, 70	syl6 35	. . . . . 6 ⊢ (¬ 𝐴 ∈ Fin → (𝑢 ∈ {𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}} → ∃𝑦 𝑢 ∈ 𝑇))
72		ax5e 1914	. . . . . 6 ⊢ (∃𝑦 𝑢 ∈ 𝑇 → 𝑢 ∈ 𝑇)
73	71, 72	syl6 35	. . . . 5 ⊢ (¬ 𝐴 ∈ Fin → (𝑢 ∈ {𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}} → 𝑢 ∈ 𝑇))
74	1, 2, 3, 73	ssrd 3939	. . . 4 ⊢ (¬ 𝐴 ∈ Fin → {𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}} ⊆ 𝑇)
75		eqid 2737	. . . . 5 ⊢ {𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}} = {𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}}
76	75	dissneq 37548	. . . 4 ⊢ (({𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}} ⊆ 𝑇 ∧ 𝑇 ∈ (TopOn‘𝐴)) → 𝑇 = 𝒫 𝐴)
77	74, 76	sylan 581	. . 3 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴)) → 𝑇 = 𝒫 𝐴)
78		nfielex 9178	. . . . 5 ⊢ (¬ 𝐴 ∈ Fin → ∃𝑦 𝑦 ∈ 𝐴)
79	78	adantr 480	. . . 4 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴)) → ∃𝑦 𝑦 ∈ 𝐴)
80		difss 4089	. . . . . . 7 ⊢ (𝐴 ∖ {𝑦}) ⊆ 𝐴
81		elfvex 6870	. . . . . . . 8 ⊢ (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ V)
82		difexg 5275	. . . . . . . 8 ⊢ (𝐴 ∈ V → (𝐴 ∖ {𝑦}) ∈ V)
83		elpwg 4558	. . . . . . . 8 ⊢ ((𝐴 ∖ {𝑦}) ∈ V → ((𝐴 ∖ {𝑦}) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ {𝑦}) ⊆ 𝐴))
84	81, 82, 83	3syl 18	. . . . . . 7 ⊢ (𝑇 ∈ (TopOn‘𝐴) → ((𝐴 ∖ {𝑦}) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ {𝑦}) ⊆ 𝐴))
85	80, 84	mpbiri 258	. . . . . 6 ⊢ (𝑇 ∈ (TopOn‘𝐴) → (𝐴 ∖ {𝑦}) ∈ 𝒫 𝐴)
86	85	adantl 481	. . . . 5 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴)) → (𝐴 ∖ {𝑦}) ∈ 𝒫 𝐴)
87		difinf 9215	. . . . . . . . . . . 12 ⊢ ((¬ 𝐴 ∈ Fin ∧ {𝑦} ∈ Fin) → ¬ (𝐴 ∖ {𝑦}) ∈ Fin)
88	17, 87	mpan2 692	. . . . . . . . . . 11 ⊢ (¬ 𝐴 ∈ Fin → ¬ (𝐴 ∖ {𝑦}) ∈ Fin)
89		0fi 8983	. . . . . . . . . . . 12 ⊢ ∅ ∈ Fin
90		eleq1 2825	. . . . . . . . . . . 12 ⊢ ((𝐴 ∖ {𝑦}) = ∅ → ((𝐴 ∖ {𝑦}) ∈ Fin ↔ ∅ ∈ Fin))
91	89, 90	mpbiri 258	. . . . . . . . . . 11 ⊢ ((𝐴 ∖ {𝑦}) = ∅ → (𝐴 ∖ {𝑦}) ∈ Fin)
92	88, 91	nsyl 140	. . . . . . . . . 10 ⊢ (¬ 𝐴 ∈ Fin → ¬ (𝐴 ∖ {𝑦}) = ∅)
93	92	ad2antrl 729	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴))) → ¬ (𝐴 ∖ {𝑦}) = ∅)
94		vsnid 4621	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ {𝑦}
95		inelcm 4418	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦 ∈ {𝑦}) → (𝐴 ∩ {𝑦}) ≠ ∅)
96	94, 95	mpan2 692	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐴 → (𝐴 ∩ {𝑦}) ≠ ∅)
97		disj4 4412	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∩ {𝑦}) = ∅ ↔ ¬ (𝐴 ∖ {𝑦}) ⊊ 𝐴)
98	97	necon2abii 2983	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ {𝑦}) ⊊ 𝐴 ↔ (𝐴 ∩ {𝑦}) ≠ ∅)
99	96, 98	sylibr 234	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐴 → (𝐴 ∖ {𝑦}) ⊊ 𝐴)
100	99	pssned 4054	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐴 → (𝐴 ∖ {𝑦}) ≠ 𝐴)
101	100	adantr 480	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴))) → (𝐴 ∖ {𝑦}) ≠ 𝐴)
102	101	neneqd 2938	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴))) → ¬ (𝐴 ∖ {𝑦}) = 𝐴)
103	93, 102	jca 511	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴))) → (¬ (𝐴 ∖ {𝑦}) = ∅ ∧ ¬ (𝐴 ∖ {𝑦}) = 𝐴))
104		pm4.56 991	. . . . . . . 8 ⊢ ((¬ (𝐴 ∖ {𝑦}) = ∅ ∧ ¬ (𝐴 ∖ {𝑦}) = 𝐴) ↔ ¬ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴))
105	103, 104	sylib 218	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴))) → ¬ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴))
106		difeq2 4073	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → (𝐴 ∖ 𝑥) = (𝐴 ∖ (𝐴 ∖ {𝑦})))
107	106	eleq1d 2822	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → ((𝐴 ∖ 𝑥) ∈ Fin ↔ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin))
108	107	notbid 318	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → (¬ (𝐴 ∖ 𝑥) ∈ Fin ↔ ¬ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin))
109		eqeq1 2741	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → (𝑥 = ∅ ↔ (𝐴 ∖ {𝑦}) = ∅))
110		eqeq1 2741	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → (𝑥 = 𝐴 ↔ (𝐴 ∖ {𝑦}) = 𝐴))
111	109, 110	orbi12d 919	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴)))
112	108, 111	orbi12d 919	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → ((¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) ↔ (¬ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin ∨ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴))))
113	112, 27	elrab2 3650	. . . . . . . . . 10 ⊢ ((𝐴 ∖ {𝑦}) ∈ 𝑇 ↔ ((𝐴 ∖ {𝑦}) ∈ 𝒫 𝐴 ∧ (¬ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin ∨ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴))))
114	85	biantrurd 532	. . . . . . . . . 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)))))
115	113, 114	bitr4id 290	. . . . . . . . 9 ⊢ (𝑇 ∈ (TopOn‘𝐴) → ((𝐴 ∖ {𝑦}) ∈ 𝑇 ↔ (¬ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin ∨ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴))))
116		dfin4 4231	. . . . . . . . . . 11 ⊢ (𝐴 ∩ {𝑦}) = (𝐴 ∖ (𝐴 ∖ {𝑦}))
117		inss2 4191	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ {𝑦}) ⊆ {𝑦}
118		ssfi 9101	. . . . . . . . . . . 12 ⊢ (({𝑦} ∈ Fin ∧ (𝐴 ∩ {𝑦}) ⊆ {𝑦}) → (𝐴 ∩ {𝑦}) ∈ Fin)
119	17, 117, 118	mp2an 693	. . . . . . . . . . 11 ⊢ (𝐴 ∩ {𝑦}) ∈ Fin
120	116, 119	eqeltrri 2834	. . . . . . . . . 10 ⊢ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin
121		biortn 938	. . . . . . . . . 10 ⊢ ((𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin → (((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴) ↔ (¬ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin ∨ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴))))
122	120, 121	ax-mp 5	. . . . . . . . 9 ⊢ (((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴) ↔ (¬ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin ∨ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴)))
123	115, 122	bitr4di 289	. . . . . . . 8 ⊢ (𝑇 ∈ (TopOn‘𝐴) → ((𝐴 ∖ {𝑦}) ∈ 𝑇 ↔ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴)))
124	123	ad2antll 730	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴))) → ((𝐴 ∖ {𝑦}) ∈ 𝑇 ↔ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴)))
125	105, 124	mtbird 325	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴))) → ¬ (𝐴 ∖ {𝑦}) ∈ 𝑇)
126	125	expcom 413	. . . . 5 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴)) → (𝑦 ∈ 𝐴 → ¬ (𝐴 ∖ {𝑦}) ∈ 𝑇))
127		nelneq2 2862	. . . . . 6 ⊢ (((𝐴 ∖ {𝑦}) ∈ 𝒫 𝐴 ∧ ¬ (𝐴 ∖ {𝑦}) ∈ 𝑇) → ¬ 𝒫 𝐴 = 𝑇)
128		eqcom 2744	. . . . . 6 ⊢ (𝑇 = 𝒫 𝐴 ↔ 𝒫 𝐴 = 𝑇)
129	127, 128	sylnibr 329	. . . . 5 ⊢ (((𝐴 ∖ {𝑦}) ∈ 𝒫 𝐴 ∧ ¬ (𝐴 ∖ {𝑦}) ∈ 𝑇) → ¬ 𝑇 = 𝒫 𝐴)
130	86, 126, 129	syl6an 685	. . . 4 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴)) → (𝑦 ∈ 𝐴 → ¬ 𝑇 = 𝒫 𝐴))
131	79, 130	exellimddv 37552	. . 3 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴)) → ¬ 𝑇 = 𝒫 𝐴)
132	77, 131	pm2.65da 817	. 2 ⊢ (¬ 𝐴 ∈ Fin → ¬ 𝑇 ∈ (TopOn‘𝐴))
133	132	con4i 114	1 ⊢ (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715   ≠ wne 2933  ∃wrex 3061  {crab 3400  Vcvv 3441  [wsbc 3741   ∖ cdif 3899   ∩ cin 3901   ⊆ wss 3902   ⊊ wpss 3903  ∅c0 4286  𝒫 cpw 4555  {csn 4581  ‘cfv 6493  Fincfn 8887  TopOnctopon 22858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-om 7811  df-1o 8399  df-en 8888  df-fin 8891  df-topgen 17367  df-top 22842  df-topon 22859

This theorem is referenced by:  topdifinffin  37555