Theorem topdifinffinlem 37330

Description: This is the core of the proof of topdifinffin 37331, 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 1912	. . . . 5 ⊢ Ⅎ𝑢 ¬ 𝐴 ∈ Fin
2		nfab1 2905	. . . . 5 ⊢ Ⅎ𝑢{𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}}
3		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑢𝑇
4		abid 2716	. . . . . . . . . . 11 ⊢ (𝑢 ∈ {𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}} ↔ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦})
5		df-rex 3069	. . . . . . . . . . 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 2735	. . . . . . . . . . . . . . 15 ⊢ {𝑦} = {𝑦}
8		vsnex 5440	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑦} ∈ V
9		snelpwi 5454	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ∈ 𝒫 𝐴)
10		eleq1 2827	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 617	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴)) → (¬ 𝐴 ∈ Fin ∧ (𝑥 = {𝑦} ∧ 𝑥 ∈ 𝒫 𝐴)))
14	13	3impb 1114	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴) → (¬ 𝐴 ∈ Fin ∧ (𝑥 = {𝑦} ∧ 𝑥 ∈ 𝒫 𝐴)))
15		3anass 1094	. . . . . . . . . . . . . . . . . . . . . . 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 9082	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ {𝑦} ∈ Fin
18		eleq1 2827	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 = {𝑦} → (𝑥 ∈ Fin ↔ {𝑦} ∈ Fin))
19	17, 18	mpbiri 258	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 = {𝑦} → 𝑥 ∈ Fin)
20		difinf 9347	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑥 ∈ Fin) → ¬ (𝐴 ∖ 𝑥) ∈ Fin)
21	19, 20	sylan2 593	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦}) → ¬ (𝐴 ∖ 𝑥) ∈ Fin)
22	21	orcd 873	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦}) → (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)))
23	22	anim2i 617	. . . . . . . . . . . . . . . . . . . . . . . 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 1109	. . . . . . . . . . . . . . . . . . . . . 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 3457	. . . . . . . . . . . . . . . . . . . . 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 2827	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = {𝑦} → (𝑥 ∈ 𝑇 ↔ {𝑦} ∈ 𝑇))
31	30	3ad2ant2 1133	. . . . . . . . . . . . . . . . . . . 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 3806	. . . . . . . . . . . . . . . . . 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 3843	. . . . . . . . . . . . . . . . . 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 3861	. . . . . . . . . . . . . . . . . 18 ⊢ ([{𝑦} / 𝑥](¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴) ↔ ([{𝑦} / 𝑥] ¬ 𝐴 ∈ Fin ∧ [{𝑦} / 𝑥]𝑥 = {𝑦} ∧ [{𝑦} / 𝑥]𝑦 ∈ 𝐴))
39		sbcg 3870	. . . . . . . . . . . . . . . . . . . 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 1156	. . . . . . . . . . . . . . . . . 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 3841	. . . . . . . . . . . . . . . . . . . 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 1157	. . . . . . . . . . . . . . . . . 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 3870	. . . . . . . . . . . . . . . . . . 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 1158	. . . . . . . . . . . . . . . . 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 3870	. . . . . . . . . . . . . . . . 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 1448	. . . . . . . . . . . . . 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 631	. . . . . . . . . . 11 ⊢ (¬ 𝐴 ∈ Fin → ((𝑦 ∈ 𝐴 ∧ 𝑢 = {𝑦}) ↔ ((𝑦 ∈ 𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑢 = {𝑦})))
57	56	exbidv 1919	. . . . . . . . . 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 1845	. . . . . . . . . 10 ⊢ (∃𝑦((𝑦 ∈ 𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑢 = {𝑦}) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ({𝑦} ∈ 𝑇 ∧ 𝑢 = {𝑦})))
61		exsimpr 1867	. . . . . . . . . 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 2827	. . . . . . . . . . 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 1845	. . . . . . . 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 1867	. . . . . . 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 1910	. . . . . 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 4000	. . . 4 ⊢ (¬ 𝐴 ∈ Fin → {𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}} ⊆ 𝑇)
75		eqid 2735	. . . . 5 ⊢ {𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}} = {𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}}
76	75	dissneq 37324	. . . 4 ⊢ (({𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}} ⊆ 𝑇 ∧ 𝑇 ∈ (TopOn‘𝐴)) → 𝑇 = 𝒫 𝐴)
77	74, 76	sylan 580	. . 3 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴)) → 𝑇 = 𝒫 𝐴)
78		nfielex 9305	. . . . 5 ⊢ (¬ 𝐴 ∈ Fin → ∃𝑦 𝑦 ∈ 𝐴)
79	78	adantr 480	. . . 4 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴)) → ∃𝑦 𝑦 ∈ 𝐴)
80		difss 4146	. . . . . . 7 ⊢ (𝐴 ∖ {𝑦}) ⊆ 𝐴
81		elfvex 6945	. . . . . . . 8 ⊢ (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ V)
82		difexg 5335	. . . . . . . 8 ⊢ (𝐴 ∈ V → (𝐴 ∖ {𝑦}) ∈ V)
83		elpwg 4608	. . . . . . . 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 9347	. . . . . . . . . . . 12 ⊢ ((¬ 𝐴 ∈ Fin ∧ {𝑦} ∈ Fin) → ¬ (𝐴 ∖ {𝑦}) ∈ Fin)
88	17, 87	mpan2 691	. . . . . . . . . . 11 ⊢ (¬ 𝐴 ∈ Fin → ¬ (𝐴 ∖ {𝑦}) ∈ Fin)
89		0fi 9081	. . . . . . . . . . . 12 ⊢ ∅ ∈ Fin
90		eleq1 2827	. . . . . . . . . . . 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 728	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴))) → ¬ (𝐴 ∖ {𝑦}) = ∅)
94		vsnid 4668	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ {𝑦}
95		inelcm 4471	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦 ∈ {𝑦}) → (𝐴 ∩ {𝑦}) ≠ ∅)
96	94, 95	mpan2 691	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐴 → (𝐴 ∩ {𝑦}) ≠ ∅)
97		disj4 4465	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∩ {𝑦}) = ∅ ↔ ¬ (𝐴 ∖ {𝑦}) ⊊ 𝐴)
98	97	necon2abii 2989	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ {𝑦}) ⊊ 𝐴 ↔ (𝐴 ∩ {𝑦}) ≠ ∅)
99	96, 98	sylibr 234	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐴 → (𝐴 ∖ {𝑦}) ⊊ 𝐴)
100	99	pssned 4111	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐴 → (𝐴 ∖ {𝑦}) ≠ 𝐴)
101	100	adantr 480	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴))) → (𝐴 ∖ {𝑦}) ≠ 𝐴)
102	101	neneqd 2943	. . . . . . . . 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		pm4.56 990	. . . . . . . 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 4130	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → (𝐴 ∖ 𝑥) = (𝐴 ∖ (𝐴 ∖ {𝑦})))
107	106	eleq1d 2824	. . . . . . . . . . . . 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 2739	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → (𝑥 = ∅ ↔ (𝐴 ∖ {𝑦}) = ∅))
110		eqeq1 2739	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → (𝑥 = 𝐴 ↔ (𝐴 ∖ {𝑦}) = 𝐴))
111	109, 110	orbi12d 918	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴)))
112	108, 111	orbi12d 918	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → ((¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) ↔ (¬ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin ∨ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴))))
113	112, 27	elrab2 3698	. . . . . . . . . 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 4284	. . . . . . . . . . 11 ⊢ (𝐴 ∩ {𝑦}) = (𝐴 ∖ (𝐴 ∖ {𝑦}))
117		inss2 4246	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ {𝑦}) ⊆ {𝑦}
118		ssfi 9212	. . . . . . . . . . . 12 ⊢ (({𝑦} ∈ Fin ∧ (𝐴 ∩ {𝑦}) ⊆ {𝑦}) → (𝐴 ∩ {𝑦}) ∈ Fin)
119	17, 117, 118	mp2an 692	. . . . . . . . . . 11 ⊢ (𝐴 ∩ {𝑦}) ∈ Fin
120	116, 119	eqeltrri 2836	. . . . . . . . . 10 ⊢ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin
121		biortn 937	. . . . . . . . . 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 729	. . . . . . 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 2864	. . . . . 6 ⊢ (((𝐴 ∖ {𝑦}) ∈ 𝒫 𝐴 ∧ ¬ (𝐴 ∖ {𝑦}) ∈ 𝑇) → ¬ 𝒫 𝐴 = 𝑇)
128		eqcom 2742	. . . . . 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 684	. . . 4 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴)) → (𝑦 ∈ 𝐴 → ¬ 𝑇 = 𝒫 𝐴))
131	79, 130	exellimddv 37328	. . 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 847   ∧ w3a 1086   = wceq 1537  ∃wex 1776   ∈ wcel 2106  {cab 2712   ≠ wne 2938  ∃wrex 3068  {crab 3433  Vcvv 3478  [wsbc 3791   ∖ cdif 3960   ∩ cin 3962   ⊆ wss 3963   ⊊ wpss 3964  ∅c0 4339  𝒫 cpw 4605  {csn 4631  ‘cfv 6563  Fincfn 8984  TopOnctopon 22932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-en 8985  df-fin 8988  df-topgen 17490  df-top 22916  df-topon 22933

This theorem is referenced by:  topdifinffin  37331