Theorem topdifinffinlem 37349

Description: This is the core of the proof of topdifinffin 37350, 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 1913	. . . . 5 ⊢ Ⅎ𝑢 ¬ 𝐴 ∈ Fin
2		nfab1 2906	. . . . 5 ⊢ Ⅎ𝑢{𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}}
3		nfcv 2904	. . . . 5 ⊢ Ⅎ𝑢𝑇
4		abid 2717	. . . . . . . . . . 11 ⊢ (𝑢 ∈ {𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}} ↔ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦})
5		df-rex 3070	. . . . . . . . . . 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 2736	. . . . . . . . . . . . . . 15 ⊢ {𝑦} = {𝑦}
8		vsnex 5433	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑦} ∈ V
9		snelpwi 5447	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ∈ 𝒫 𝐴)
10		eleq1 2828	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 9084	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ {𝑦} ∈ Fin
18		eleq1 2828	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 = {𝑦} → (𝑥 ∈ Fin ↔ {𝑦} ∈ Fin))
19	17, 18	mpbiri 258	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 = {𝑦} → 𝑥 ∈ Fin)
20		difinf 9350	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 3459	. . . . . . . . . . . . . . . . . . . . 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 2828	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = {𝑦} → (𝑥 ∈ 𝑇 ↔ {𝑦} ∈ 𝑇))
31	30	3ad2ant2 1134	. . . . . . . . . . . . . . . . . . . 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 3802	. . . . . . . . . . . . . . . . . 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 3836	. . . . . . . . . . . . . . . . . 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 3854	. . . . . . . . . . . . . . . . . 18 ⊢ ([{𝑦} / 𝑥](¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴) ↔ ([{𝑦} / 𝑥] ¬ 𝐴 ∈ Fin ∧ [{𝑦} / 𝑥]𝑥 = {𝑦} ∧ [{𝑦} / 𝑥]𝑦 ∈ 𝐴))
39		sbcg 3862	. . . . . . . . . . . . . . . . . . . 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 1157	. . . . . . . . . . . . . . . . . 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 3834	. . . . . . . . . . . . . . . . . . . 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 1158	. . . . . . . . . . . . . . . . . 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 3862	. . . . . . . . . . . . . . . . . . 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 1159	. . . . . . . . . . . . . . . . 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 3862	. . . . . . . . . . . . . . . . 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 1450	. . . . . . . . . . . . . 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 1920	. . . . . . . . . 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 1847	. . . . . . . . . 10 ⊢ (∃𝑦((𝑦 ∈ 𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑢 = {𝑦}) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ({𝑦} ∈ 𝑇 ∧ 𝑢 = {𝑦})))
61		exsimpr 1868	. . . . . . . . . 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 2828	. . . . . . . . . . 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 1847	. . . . . . . 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 1868	. . . . . . 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 1911	. . . . . 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 3987	. . . 4 ⊢ (¬ 𝐴 ∈ Fin → {𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}} ⊆ 𝑇)
75		eqid 2736	. . . . 5 ⊢ {𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}} = {𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}}
76	75	dissneq 37343	. . . 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 9308	. . . . 5 ⊢ (¬ 𝐴 ∈ Fin → ∃𝑦 𝑦 ∈ 𝐴)
79	78	adantr 480	. . . 4 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴)) → ∃𝑦 𝑦 ∈ 𝐴)
80		difss 4135	. . . . . . 7 ⊢ (𝐴 ∖ {𝑦}) ⊆ 𝐴
81		elfvex 6943	. . . . . . . 8 ⊢ (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ V)
82		difexg 5328	. . . . . . . 8 ⊢ (𝐴 ∈ V → (𝐴 ∖ {𝑦}) ∈ V)
83		elpwg 4602	. . . . . . . 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 9350	. . . . . . . . . . . 12 ⊢ ((¬ 𝐴 ∈ Fin ∧ {𝑦} ∈ Fin) → ¬ (𝐴 ∖ {𝑦}) ∈ Fin)
88	17, 87	mpan2 691	. . . . . . . . . . 11 ⊢ (¬ 𝐴 ∈ Fin → ¬ (𝐴 ∖ {𝑦}) ∈ Fin)
89		0fi 9083	. . . . . . . . . . . 12 ⊢ ∅ ∈ Fin
90		eleq1 2828	. . . . . . . . . . . 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 4662	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ {𝑦}
95		inelcm 4464	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦 ∈ {𝑦}) → (𝐴 ∩ {𝑦}) ≠ ∅)
96	94, 95	mpan2 691	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐴 → (𝐴 ∩ {𝑦}) ≠ ∅)
97		disj4 4458	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∩ {𝑦}) = ∅ ↔ ¬ (𝐴 ∖ {𝑦}) ⊊ 𝐴)
98	97	necon2abii 2990	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ {𝑦}) ⊊ 𝐴 ↔ (𝐴 ∩ {𝑦}) ≠ ∅)
99	96, 98	sylibr 234	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐴 → (𝐴 ∖ {𝑦}) ⊊ 𝐴)
100	99	pssned 4100	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐴 → (𝐴 ∖ {𝑦}) ≠ 𝐴)
101	100	adantr 480	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴))) → (𝐴 ∖ {𝑦}) ≠ 𝐴)
102	101	neneqd 2944	. . . . . . . . 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 4119	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → (𝐴 ∖ 𝑥) = (𝐴 ∖ (𝐴 ∖ {𝑦})))
107	106	eleq1d 2825	. . . . . . . . . . . . 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 2740	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → (𝑥 = ∅ ↔ (𝐴 ∖ {𝑦}) = ∅))
110		eqeq1 2740	. . . . . . . . . . . . 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 3694	. . . . . . . . . 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 4277	. . . . . . . . . . 11 ⊢ (𝐴 ∩ {𝑦}) = (𝐴 ∖ (𝐴 ∖ {𝑦}))
117		inss2 4237	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ {𝑦}) ⊆ {𝑦}
118		ssfi 9214	. . . . . . . . . . . 12 ⊢ (({𝑦} ∈ Fin ∧ (𝐴 ∩ {𝑦}) ⊆ {𝑦}) → (𝐴 ∩ {𝑦}) ∈ Fin)
119	17, 117, 118	mp2an 692	. . . . . . . . . . 11 ⊢ (𝐴 ∩ {𝑦}) ∈ Fin
120	116, 119	eqeltrri 2837	. . . . . . . . . 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 2865	. . . . . 6 ⊢ (((𝐴 ∖ {𝑦}) ∈ 𝒫 𝐴 ∧ ¬ (𝐴 ∖ {𝑦}) ∈ 𝑇) → ¬ 𝒫 𝐴 = 𝑇)
128		eqcom 2743	. . . . . 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 37347	. . 3 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴)) → ¬ 𝑇 = 𝒫 𝐴)
132	77, 131	pm2.65da 816	. 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 1539  ∃wex 1778   ∈ wcel 2107  {cab 2713   ≠ wne 2939  ∃wrex 3069  {crab 3435  Vcvv 3479  [wsbc 3787   ∖ cdif 3947   ∩ cin 3949   ⊆ wss 3950   ⊊ wpss 3951  ∅c0 4332  𝒫 cpw 4599  {csn 4625  ‘cfv 6560  Fincfn 8986  TopOnctopon 22917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-om 7889  df-1o 8507  df-en 8987  df-fin 8990  df-topgen 17489  df-top 22901  df-topon 22918

This theorem is referenced by:  topdifinffin  37350