Theorem topdifinffinlem 37846

Description: This is the core of the proof of topdifinffin 37847, 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 1936	. . . . 5 ⊢ Ⅎ𝑢 ¬ 𝐴 ∈ Fin
2		nfab1 2928	. . . . 5 ⊢ Ⅎ𝑢{𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}}
3		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑢𝑇
4		abid 2746	. . . . . . . . . . 11 ⊢ (𝑢 ∈ {𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}} ↔ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦})
5		df-rex 3089	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝐴 𝑢 = {𝑦} ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑢 = {𝑦}))
6	4, 5	bitri 277	. . . . . . . . . 10 ⊢ (𝑢 ∈ {𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}} ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑢 = {𝑦}))
7		eqid 2764	. . . . . . . . . . . . . . 15 ⊢ {𝑦} = {𝑦}
8		vsnex 5394	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑦} ∈ V
9		snelpwi 5413	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ∈ 𝒫 𝐴)
10		eleq1 2852	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 = {𝑦} → (𝑥 ∈ 𝒫 𝐴 ↔ {𝑦} ∈ 𝒫 𝐴))
11	9, 10	imbitrrid 248	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = {𝑦} → (𝑦 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴))
12	11	imdistani 576	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴) → (𝑥 = {𝑦} ∧ 𝑥 ∈ 𝒫 𝐴))
13	12	anim2i 626	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴)) → (¬ 𝐴 ∈ Fin ∧ (𝑥 = {𝑦} ∧ 𝑥 ∈ 𝒫 𝐴)))
14	13	3impb 1128	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴) → (¬ 𝐴 ∈ Fin ∧ (𝑥 = {𝑦} ∧ 𝑥 ∈ 𝒫 𝐴)))
15		3anass 1107	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑥 ∈ 𝒫 𝐴) ↔ (¬ 𝐴 ∈ Fin ∧ (𝑥 = {𝑦} ∧ 𝑥 ∈ 𝒫 𝐴)))
16	14, 15	sylibr 236	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴) → (¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑥 ∈ 𝒫 𝐴))
17		snfi 9026	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ {𝑦} ∈ Fin
18		eleq1 2852	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 = {𝑦} → (𝑥 ∈ Fin ↔ {𝑦} ∈ Fin))
19	17, 18	mpbiri 260	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 = {𝑦} → 𝑥 ∈ Fin)
20		difinf 9257	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑥 ∈ Fin) → ¬ (𝐴 ∖ 𝑥) ∈ Fin)
21	19, 20	sylan2 602	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦}) → ¬ (𝐴 ∖ 𝑥) ∈ Fin)
22	21	orcd 884	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦}) → (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)))
23	22	anim2i 626	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦})) → (𝑥 ∈ 𝒫 𝐴 ∧ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
24	23	ancoms 462	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦}) ∧ 𝑥 ∈ 𝒫 𝐴) → (𝑥 ∈ 𝒫 𝐴 ∧ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
25	24	3impa 1123	. . . . . . . . . . . . . . . . . . . . . 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 3439	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ 𝑇 ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))))
29	26, 28	sylibr 236	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝑇)
30		eleq1 2852	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = {𝑦} → (𝑥 ∈ 𝑇 ↔ {𝑦} ∈ 𝑇))
31	30	3ad2ant2 1148	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ 𝑇 ↔ {𝑦} ∈ 𝑇))
32	29, 31	mpbid 234	. . . . . . . . . . . . . . . . . . 19 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴) → {𝑦} ∈ 𝑇)
33	32	sbcth 3761	. . . . . . . . . . . . . . . . . 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 3794	. . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . 16 ⊢ ([{𝑦} / 𝑥](¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴) → [{𝑦} / 𝑥]{𝑦} ∈ 𝑇)
38		sbc3an 3810	. . . . . . . . . . . . . . . . . 18 ⊢ ([{𝑦} / 𝑥](¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴) ↔ ([{𝑦} / 𝑥] ¬ 𝐴 ∈ Fin ∧ [{𝑦} / 𝑥]𝑥 = {𝑦} ∧ [{𝑦} / 𝑥]𝑦 ∈ 𝐴))
39		sbcg 3818	. . . . . . . . . . . . . . . . . . . 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 1171	. . . . . . . . . . . . . . . . . 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 3792	. . . . . . . . . . . . . . . . . . . 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 1172	. . . . . . . . . . . . . . . . . 18 ⊢ ((¬ 𝐴 ∈ Fin ∧ [{𝑦} / 𝑥]𝑥 = {𝑦} ∧ [{𝑦} / 𝑥]𝑦 ∈ 𝐴) ↔ (¬ 𝐴 ∈ Fin ∧ {𝑦} = {𝑦} ∧ [{𝑦} / 𝑥]𝑦 ∈ 𝐴))
45	38, 41, 44	3bitri 299	. . . . . . . . . . . . . . . . 17 ⊢ ([{𝑦} / 𝑥](¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴) ↔ (¬ 𝐴 ∈ Fin ∧ {𝑦} = {𝑦} ∧ [{𝑦} / 𝑥]𝑦 ∈ 𝐴))
46		sbcg 3818	. . . . . . . . . . . . . . . . . . 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 1173	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ 𝐴 ∈ Fin ∧ {𝑦} = {𝑦} ∧ [{𝑦} / 𝑥]𝑦 ∈ 𝐴) ↔ (¬ 𝐴 ∈ Fin ∧ {𝑦} = {𝑦} ∧ 𝑦 ∈ 𝐴))
49	45, 48	bitri 277	. . . . . . . . . . . . . . . 16 ⊢ ([{𝑦} / 𝑥](¬ 𝐴 ∈ Fin ∧ 𝑥 = {𝑦} ∧ 𝑦 ∈ 𝐴) ↔ (¬ 𝐴 ∈ Fin ∧ {𝑦} = {𝑦} ∧ 𝑦 ∈ 𝐴))
50		sbcg 3818	. . . . . . . . . . . . . . . . 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 293	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝐴 ∈ Fin ∧ {𝑦} = {𝑦} ∧ 𝑦 ∈ 𝐴) → {𝑦} ∈ 𝑇)
53	7, 52	mp3an2 1472	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑦 ∈ 𝐴) → {𝑦} ∈ 𝑇)
54	53	ex 416	. . . . . . . . . . . . 13 ⊢ (¬ 𝐴 ∈ Fin → (𝑦 ∈ 𝐴 → {𝑦} ∈ 𝑇))
55	54	pm4.71d 569	. . . . . . . . . . . 12 ⊢ (¬ 𝐴 ∈ Fin → (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ 𝐴 ∧ {𝑦} ∈ 𝑇)))
56	55	anbi1d 640	. . . . . . . . . . 11 ⊢ (¬ 𝐴 ∈ Fin → ((𝑦 ∈ 𝐴 ∧ 𝑢 = {𝑦}) ↔ ((𝑦 ∈ 𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑢 = {𝑦})))
57	56	exbidv 1943	. . . . . . . . . 10 ⊢ (¬ 𝐴 ∈ Fin → (∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑢 = {𝑦}) ↔ ∃𝑦((𝑦 ∈ 𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑢 = {𝑦})))
58	6, 57	bitrid 285	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ Fin → (𝑢 ∈ {𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}} ↔ ∃𝑦((𝑦 ∈ 𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑢 = {𝑦})))
59		anass 472	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ 𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑢 = {𝑦}) ↔ (𝑦 ∈ 𝐴 ∧ ({𝑦} ∈ 𝑇 ∧ 𝑢 = {𝑦})))
60	59	exbii 1870	. . . . . . . . . 10 ⊢ (∃𝑦((𝑦 ∈ 𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑢 = {𝑦}) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ({𝑦} ∈ 𝑇 ∧ 𝑢 = {𝑦})))
61		exsimpr 1891	. . . . . . . . . 10 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ ({𝑦} ∈ 𝑇 ∧ 𝑢 = {𝑦})) → ∃𝑦({𝑦} ∈ 𝑇 ∧ 𝑢 = {𝑦}))
62	60, 61	sylbi 219	. . . . . . . . 9 ⊢ (∃𝑦((𝑦 ∈ 𝐴 ∧ {𝑦} ∈ 𝑇) ∧ 𝑢 = {𝑦}) → ∃𝑦({𝑦} ∈ 𝑇 ∧ 𝑢 = {𝑦}))
63	58, 62	biimtrdi 255	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ Fin → (𝑢 ∈ {𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}} → ∃𝑦({𝑦} ∈ 𝑇 ∧ 𝑢 = {𝑦})))
64		ancom 464	. . . . . . . . . 10 ⊢ (({𝑦} ∈ 𝑇 ∧ 𝑢 = {𝑦}) ↔ (𝑢 = {𝑦} ∧ {𝑦} ∈ 𝑇))
65		eleq1 2852	. . . . . . . . . . 11 ⊢ (𝑢 = {𝑦} → (𝑢 ∈ 𝑇 ↔ {𝑦} ∈ 𝑇))
66	65	pm5.32i 582	. . . . . . . . . 10 ⊢ ((𝑢 = {𝑦} ∧ 𝑢 ∈ 𝑇) ↔ (𝑢 = {𝑦} ∧ {𝑦} ∈ 𝑇))
67	64, 66	bitr4i 280	. . . . . . . . 9 ⊢ (({𝑦} ∈ 𝑇 ∧ 𝑢 = {𝑦}) ↔ (𝑢 = {𝑦} ∧ 𝑢 ∈ 𝑇))
68	67	exbii 1870	. . . . . . . 8 ⊢ (∃𝑦({𝑦} ∈ 𝑇 ∧ 𝑢 = {𝑦}) ↔ ∃𝑦(𝑢 = {𝑦} ∧ 𝑢 ∈ 𝑇))
69	63, 68	imbitrdi 253	. . . . . . 7 ⊢ (¬ 𝐴 ∈ Fin → (𝑢 ∈ {𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}} → ∃𝑦(𝑢 = {𝑦} ∧ 𝑢 ∈ 𝑇)))
70		exsimpr 1891	. . . . . . 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 1934	. . . . . 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 3943	. . . 4 ⊢ (¬ 𝐴 ∈ Fin → {𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}} ⊆ 𝑇)
75		eqid 2764	. . . . 5 ⊢ {𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}} = {𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}}
76	75	dissneq 37840	. . . 4 ⊢ (({𝑢 ∣ ∃𝑦 ∈ 𝐴 𝑢 = {𝑦}} ⊆ 𝑇 ∧ 𝑇 ∈ (TopOn‘𝐴)) → 𝑇 = 𝒫 𝐴)
77	74, 76	sylan 589	. . 3 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴)) → 𝑇 = 𝒫 𝐴)
78		nfielex 9220	. . . . 5 ⊢ (¬ 𝐴 ∈ Fin → ∃𝑦 𝑦 ∈ 𝐴)
79	78	adantr 484	. . . 4 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴)) → ∃𝑦 𝑦 ∈ 𝐴)
80		difss 4091	. . . . . . 7 ⊢ (𝐴 ∖ {𝑦}) ⊆ 𝐴
81		elfvex 6904	. . . . . . . 8 ⊢ (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ V)
82		difexg 5287	. . . . . . . 8 ⊢ (𝐴 ∈ V → (𝐴 ∖ {𝑦}) ∈ V)
83		elpwg 4560	. . . . . . . 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 260	. . . . . 6 ⊢ (𝑇 ∈ (TopOn‘𝐴) → (𝐴 ∖ {𝑦}) ∈ 𝒫 𝐴)
86	85	adantl 485	. . . . 5 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴)) → (𝐴 ∖ {𝑦}) ∈ 𝒫 𝐴)
87		difinf 9257	. . . . . . . . . . . 12 ⊢ ((¬ 𝐴 ∈ Fin ∧ {𝑦} ∈ Fin) → ¬ (𝐴 ∖ {𝑦}) ∈ Fin)
88	17, 87	mpan2 701	. . . . . . . . . . 11 ⊢ (¬ 𝐴 ∈ Fin → ¬ (𝐴 ∖ {𝑦}) ∈ Fin)
89		0fi 9025	. . . . . . . . . . . 12 ⊢ ∅ ∈ Fin
90		eleq1 2852	. . . . . . . . . . . 12 ⊢ ((𝐴 ∖ {𝑦}) = ∅ → ((𝐴 ∖ {𝑦}) ∈ Fin ↔ ∅ ∈ Fin))
91	89, 90	mpbiri 260	. . . . . . . . . . 11 ⊢ ((𝐴 ∖ {𝑦}) = ∅ → (𝐴 ∖ {𝑦}) ∈ Fin)
92	88, 91	nsyl 140	. . . . . . . . . 10 ⊢ (¬ 𝐴 ∈ Fin → ¬ (𝐴 ∖ {𝑦}) = ∅)
93	92	ad2antrl 738	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴))) → ¬ (𝐴 ∖ {𝑦}) = ∅)
94		vsnid 4624	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ {𝑦}
95		inelcm 4421	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦 ∈ {𝑦}) → (𝐴 ∩ {𝑦}) ≠ ∅)
96	94, 95	mpan2 701	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐴 → (𝐴 ∩ {𝑦}) ≠ ∅)
97		disj4 4415	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∩ {𝑦}) = ∅ ↔ ¬ (𝐴 ∖ {𝑦}) ⊊ 𝐴)
98	97	necon2abii 3009	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ {𝑦}) ⊊ 𝐴 ↔ (𝐴 ∩ {𝑦}) ≠ ∅)
99	96, 98	sylibr 236	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐴 → (𝐴 ∖ {𝑦}) ⊊ 𝐴)
100	99	pssned 4056	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐴 → (𝐴 ∖ {𝑦}) ≠ 𝐴)
101	100	adantr 484	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴))) → (𝐴 ∖ {𝑦}) ≠ 𝐴)
102	101	neneqd 2964	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴))) → ¬ (𝐴 ∖ {𝑦}) = 𝐴)
103	93, 102	jca 519	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ (¬ 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))
105	103, 104	sylib 220	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴))) → ¬ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴))
106		difeq2 4076	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → (𝐴 ∖ 𝑥) = (𝐴 ∖ (𝐴 ∖ {𝑦})))
107	106	eleq1d 2849	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → ((𝐴 ∖ 𝑥) ∈ Fin ↔ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin))
108	107	notbid 320	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → (¬ (𝐴 ∖ 𝑥) ∈ Fin ↔ ¬ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin))
109		eqeq1 2768	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → (𝑥 = ∅ ↔ (𝐴 ∖ {𝑦}) = ∅))
110		eqeq1 2768	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → (𝑥 = 𝐴 ↔ (𝐴 ∖ {𝑦}) = 𝐴))
111	109, 110	orbi12d 929	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → ((𝑥 = ∅ ∨ 𝑥 = 𝐴) ↔ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴)))
112	108, 111	orbi12d 929	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → ((¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴)) ↔ (¬ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin ∨ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴))))
113	112, 27	elrab2 3656	. . . . . . . . . 10 ⊢ ((𝐴 ∖ {𝑦}) ∈ 𝑇 ↔ ((𝐴 ∖ {𝑦}) ∈ 𝒫 𝐴 ∧ (¬ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin ∨ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴))))
114	85	biantrurd 540	. . . . . . . . . 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 292	. . . . . . . . 9 ⊢ (𝑇 ∈ (TopOn‘𝐴) → ((𝐴 ∖ {𝑦}) ∈ 𝑇 ↔ (¬ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin ∨ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴))))
116		dfin4 4232	. . . . . . . . . . 11 ⊢ (𝐴 ∩ {𝑦}) = (𝐴 ∖ (𝐴 ∖ {𝑦}))
117		inss2 4191	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ {𝑦}) ⊆ {𝑦}
118		ssfi 9143	. . . . . . . . . . . 12 ⊢ (({𝑦} ∈ Fin ∧ (𝐴 ∩ {𝑦}) ⊆ {𝑦}) → (𝐴 ∩ {𝑦}) ∈ Fin)
119	17, 117, 118	mp2an 702	. . . . . . . . . . 11 ⊢ (𝐴 ∩ {𝑦}) ∈ Fin
120	116, 119	eqeltrri 2861	. . . . . . . . . 10 ⊢ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin
121		biortn 948	. . . . . . . . . 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 291	. . . . . . . 8 ⊢ (𝑇 ∈ (TopOn‘𝐴) → ((𝐴 ∖ {𝑦}) ∈ 𝑇 ↔ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴)))
124	123	ad2antll 739	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴))) → ((𝐴 ∖ {𝑦}) ∈ 𝑇 ↔ ((𝐴 ∖ {𝑦}) = ∅ ∨ (𝐴 ∖ {𝑦}) = 𝐴)))
125	105, 124	mtbird 327	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ (¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴))) → ¬ (𝐴 ∖ {𝑦}) ∈ 𝑇)
126	125	expcom 417	. . . . 5 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴)) → (𝑦 ∈ 𝐴 → ¬ (𝐴 ∖ {𝑦}) ∈ 𝑇))
127		nelneq2 2889	. . . . . 6 ⊢ (((𝐴 ∖ {𝑦}) ∈ 𝒫 𝐴 ∧ ¬ (𝐴 ∖ {𝑦}) ∈ 𝑇) → ¬ 𝒫 𝐴 = 𝑇)
128		eqcom 2771	. . . . . 6 ⊢ (𝑇 = 𝒫 𝐴 ↔ 𝒫 𝐴 = 𝑇)
129	127, 128	sylnibr 331	. . . . 5 ⊢ (((𝐴 ∖ {𝑦}) ∈ 𝒫 𝐴 ∧ ¬ (𝐴 ∖ {𝑦}) ∈ 𝑇) → ¬ 𝑇 = 𝒫 𝐴)
130	86, 126, 129	syl6an 694	. . . 4 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴)) → (𝑦 ∈ 𝐴 → ¬ 𝑇 = 𝒫 𝐴))
131	79, 130	exellimddv 37844	. . 3 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝑇 ∈ (TopOn‘𝐴)) → ¬ 𝑇 = 𝒫 𝐴)
132	77, 131	pm2.65da 826	. 2 ⊢ (¬ 𝐴 ∈ Fin → ¬ 𝑇 ∈ (TopOn‘𝐴))
133	132	con4i 114	1 ⊢ (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   ∧ w3a 1099   = wceq 1562  ∃wex 1801   ∈ wcel 2144  {cab 2742   ≠ wne 2959  ∃wrex 3088  {crab 3416  Vcvv 3456  [wsbc 3746   ∖ cdif 3903   ∩ cin 3905   ⊆ wss 3906   ⊊ wpss 3907  ∅c0 4287  𝒫 cpw 4557  {csn 4584  ‘cfv 6523  Fincfn 8929  TopOnctopon 22972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-om 7849  df-1o 8439  df-en 8930  df-fin 8933  df-topgen 17474  df-top 22956  df-topon 22973

This theorem is referenced by:  topdifinffin  37847