Theorem filconn 21687

Description: A filter gives rise to a connected topology. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
filconn	⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ {∅}) ∈ Conn)

Proof of Theorem filconn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
2		filunibas 21685	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋)
3	2	fveq2d 6195	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → (Fil‘∪ 𝐹) = (Fil‘𝑋))
4	1, 3	eleqtrrd 2704	. 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (Fil‘∪ 𝐹))
5		nss 3663	. . . . . . . 8 ⊢ (¬ 𝑥 ⊆ {∅} ↔ ∃𝑦(𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ {∅}))
6		simpll 790	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ {∅})) → 𝐹 ∈ (Fil‘∪ 𝐹))
7		ssel2 3598	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ⊆ (𝐹 ∪ {∅}) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ (𝐹 ∪ {∅}))
8	7	adantll 750	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ (𝐹 ∪ {∅}))
9		elun 3753	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (𝐹 ∪ {∅}) ↔ (𝑦 ∈ 𝐹 ∨ 𝑦 ∈ {∅}))
10	8, 9	sylib 208	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ 𝑦 ∈ 𝑥) → (𝑦 ∈ 𝐹 ∨ 𝑦 ∈ {∅}))
11	10	orcomd 403	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ 𝑦 ∈ 𝑥) → (𝑦 ∈ {∅} ∨ 𝑦 ∈ 𝐹))
12	11	ord 392	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ 𝑦 ∈ 𝑥) → (¬ 𝑦 ∈ {∅} → 𝑦 ∈ 𝐹))
13	12	impr 649	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ {∅})) → 𝑦 ∈ 𝐹)
14		uniss 4458	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ (𝐹 ∪ {∅}) → ∪ 𝑥 ⊆ ∪ (𝐹 ∪ {∅}))
15	14	ad2antlr 763	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ {∅})) → ∪ 𝑥 ⊆ ∪ (𝐹 ∪ {∅}))
16		uniun 4456	. . . . . . . . . . . . . 14 ⊢ ∪ (𝐹 ∪ {∅}) = (∪ 𝐹 ∪ ∪ {∅})
17		0ex 4790	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ V
18	17	unisn 4451	. . . . . . . . . . . . . . 15 ⊢ ∪ {∅} = ∅
19	18	uneq2i 3764	. . . . . . . . . . . . . 14 ⊢ (∪ 𝐹 ∪ ∪ {∅}) = (∪ 𝐹 ∪ ∅)
20		un0 3967	. . . . . . . . . . . . . 14 ⊢ (∪ 𝐹 ∪ ∅) = ∪ 𝐹
21	16, 19, 20	3eqtrri 2649	. . . . . . . . . . . . 13 ⊢ ∪ 𝐹 = ∪ (𝐹 ∪ {∅})
22	15, 21	syl6sseqr 3652	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ {∅})) → ∪ 𝑥 ⊆ ∪ 𝐹)
23		elssuni 4467	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑥 → 𝑦 ⊆ ∪ 𝑥)
24	23	ad2antrl 764	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ {∅})) → 𝑦 ⊆ ∪ 𝑥)
25		filss 21657	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ (𝑦 ∈ 𝐹 ∧ ∪ 𝑥 ⊆ ∪ 𝐹 ∧ 𝑦 ⊆ ∪ 𝑥)) → ∪ 𝑥 ∈ 𝐹)
26	6, 13, 22, 24, 25	syl13anc 1328	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ {∅})) → ∪ 𝑥 ∈ 𝐹)
27		elun1 3780	. . . . . . . . . . 11 ⊢ (∪ 𝑥 ∈ 𝐹 → ∪ 𝑥 ∈ (𝐹 ∪ {∅}))
28	26, 27	syl 17	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ {∅})) → ∪ 𝑥 ∈ (𝐹 ∪ {∅}))
29	28	ex 450	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) → ((𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ {∅}) → ∪ 𝑥 ∈ (𝐹 ∪ {∅})))
30	29	exlimdv 1861	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) → (∃𝑦(𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ {∅}) → ∪ 𝑥 ∈ (𝐹 ∪ {∅})))
31	5, 30	syl5bi 232	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) → (¬ 𝑥 ⊆ {∅} → ∪ 𝑥 ∈ (𝐹 ∪ {∅})))
32		uni0b 4463	. . . . . . . 8 ⊢ (∪ 𝑥 = ∅ ↔ 𝑥 ⊆ {∅})
33		ssun2 3777	. . . . . . . . . 10 ⊢ {∅} ⊆ (𝐹 ∪ {∅})
34	17	snid 4208	. . . . . . . . . 10 ⊢ ∅ ∈ {∅}
35	33, 34	sselii 3600	. . . . . . . . 9 ⊢ ∅ ∈ (𝐹 ∪ {∅})
36		eleq1 2689	. . . . . . . . 9 ⊢ (∪ 𝑥 = ∅ → (∪ 𝑥 ∈ (𝐹 ∪ {∅}) ↔ ∅ ∈ (𝐹 ∪ {∅})))
37	35, 36	mpbiri 248	. . . . . . . 8 ⊢ (∪ 𝑥 = ∅ → ∪ 𝑥 ∈ (𝐹 ∪ {∅}))
38	32, 37	sylbir 225	. . . . . . 7 ⊢ (𝑥 ⊆ {∅} → ∪ 𝑥 ∈ (𝐹 ∪ {∅}))
39	31, 38	pm2.61d2 172	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) → ∪ 𝑥 ∈ (𝐹 ∪ {∅}))
40	39	ex 450	. . . . 5 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → (𝑥 ⊆ (𝐹 ∪ {∅}) → ∪ 𝑥 ∈ (𝐹 ∪ {∅})))
41	40	alrimiv 1855	. . . 4 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → ∀𝑥(𝑥 ⊆ (𝐹 ∪ {∅}) → ∪ 𝑥 ∈ (𝐹 ∪ {∅})))
42		filin 21658	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 ∩ 𝑦) ∈ 𝐹)
43		elun1 3780	. . . . . . . . . 10 ⊢ ((𝑥 ∩ 𝑦) ∈ 𝐹 → (𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
44	42, 43	syl 17	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
45	44	3expa 1265	. . . . . . . 8 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ 𝐹) ∧ 𝑦 ∈ 𝐹) → (𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
46	45	ralrimiva 2966	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ 𝐹) → ∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
47		elsni 4194	. . . . . . . . 9 ⊢ (𝑦 ∈ {∅} → 𝑦 = ∅)
48		ineq2 3808	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → (𝑥 ∩ 𝑦) = (𝑥 ∩ ∅))
49		in0 3968	. . . . . . . . . . 11 ⊢ (𝑥 ∩ ∅) = ∅
50	48, 49	syl6eq 2672	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → (𝑥 ∩ 𝑦) = ∅)
51	50, 35	syl6eqel 2709	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
52	47, 51	syl 17	. . . . . . . 8 ⊢ (𝑦 ∈ {∅} → (𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
53	52	rgen 2922	. . . . . . 7 ⊢ ∀𝑦 ∈ {∅} (𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅})
54		ralun 3795	. . . . . . 7 ⊢ ((∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}) ∧ ∀𝑦 ∈ {∅} (𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅})) → ∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
55	46, 53, 54	sylancl 694	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ 𝐹) → ∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
56	55	ralrimiva 2966	. . . . 5 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
57		elsni 4194	. . . . . . 7 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅)
58		ineq1 3807	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (𝑥 ∩ 𝑦) = (∅ ∩ 𝑦))
59		0in 3969	. . . . . . . . . 10 ⊢ (∅ ∩ 𝑦) = ∅
60	58, 59	syl6eq 2672	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑥 ∩ 𝑦) = ∅)
61	60, 35	syl6eqel 2709	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
62	61	ralrimivw 2967	. . . . . . 7 ⊢ (𝑥 = ∅ → ∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
63	57, 62	syl 17	. . . . . 6 ⊢ (𝑥 ∈ {∅} → ∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
64	63	rgen 2922	. . . . 5 ⊢ ∀𝑥 ∈ {∅}∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅})
65		ralun 3795	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐹 ∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}) ∧ ∀𝑥 ∈ {∅}∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅})) → ∀𝑥 ∈ (𝐹 ∪ {∅})∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
66	56, 64, 65	sylancl 694	. . . 4 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → ∀𝑥 ∈ (𝐹 ∪ {∅})∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
67		p0ex 4853	. . . . . 6 ⊢ {∅} ∈ V
68		unexg 6959	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ {∅} ∈ V) → (𝐹 ∪ {∅}) ∈ V)
69	67, 68	mpan2 707	. . . . 5 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → (𝐹 ∪ {∅}) ∈ V)
70		istopg 20700	. . . . 5 ⊢ ((𝐹 ∪ {∅}) ∈ V → ((𝐹 ∪ {∅}) ∈ Top ↔ (∀𝑥(𝑥 ⊆ (𝐹 ∪ {∅}) → ∪ 𝑥 ∈ (𝐹 ∪ {∅})) ∧ ∀𝑥 ∈ (𝐹 ∪ {∅})∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))))
71	69, 70	syl 17	. . . 4 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → ((𝐹 ∪ {∅}) ∈ Top ↔ (∀𝑥(𝑥 ⊆ (𝐹 ∪ {∅}) → ∪ 𝑥 ∈ (𝐹 ∪ {∅})) ∧ ∀𝑥 ∈ (𝐹 ∪ {∅})∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))))
72	41, 66, 71	mpbir2and 957	. . 3 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → (𝐹 ∪ {∅}) ∈ Top)
73	21	cldopn 20835	. . . . . . . 8 ⊢ (𝑥 ∈ (Clsd‘(𝐹 ∪ {∅})) → (∪ 𝐹 ∖ 𝑥) ∈ (𝐹 ∪ {∅}))
74		elun 3753	. . . . . . . 8 ⊢ ((∪ 𝐹 ∖ 𝑥) ∈ (𝐹 ∪ {∅}) ↔ ((∪ 𝐹 ∖ 𝑥) ∈ 𝐹 ∨ (∪ 𝐹 ∖ 𝑥) ∈ {∅}))
75	73, 74	sylib 208	. . . . . . 7 ⊢ (𝑥 ∈ (Clsd‘(𝐹 ∪ {∅})) → ((∪ 𝐹 ∖ 𝑥) ∈ 𝐹 ∨ (∪ 𝐹 ∖ 𝑥) ∈ {∅}))
76		elun 3753	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐹 ∪ {∅}) ↔ (𝑥 ∈ 𝐹 ∨ 𝑥 ∈ {∅}))
77		filfbas 21652	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → 𝐹 ∈ (fBas‘∪ 𝐹))
78		fbncp 21643	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ (fBas‘∪ 𝐹) ∧ 𝑥 ∈ 𝐹) → ¬ (∪ 𝐹 ∖ 𝑥) ∈ 𝐹)
79	77, 78	sylan 488	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ 𝐹) → ¬ (∪ 𝐹 ∖ 𝑥) ∈ 𝐹)
80	79	pm2.21d 118	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ 𝐹) → ((∪ 𝐹 ∖ 𝑥) ∈ 𝐹 → 𝑥 = ∅))
81	80	ex 450	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → (𝑥 ∈ 𝐹 → ((∪ 𝐹 ∖ 𝑥) ∈ 𝐹 → 𝑥 = ∅)))
82	57	a1i13 27	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → (𝑥 ∈ {∅} → ((∪ 𝐹 ∖ 𝑥) ∈ 𝐹 → 𝑥 = ∅)))
83	81, 82	jaod 395	. . . . . . . . . 10 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → ((𝑥 ∈ 𝐹 ∨ 𝑥 ∈ {∅}) → ((∪ 𝐹 ∖ 𝑥) ∈ 𝐹 → 𝑥 = ∅)))
84	76, 83	syl5bi 232	. . . . . . . . 9 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → (𝑥 ∈ (𝐹 ∪ {∅}) → ((∪ 𝐹 ∖ 𝑥) ∈ 𝐹 → 𝑥 = ∅)))
85	84	imp 445	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → ((∪ 𝐹 ∖ 𝑥) ∈ 𝐹 → 𝑥 = ∅))
86		elsni 4194	. . . . . . . . 9 ⊢ ((∪ 𝐹 ∖ 𝑥) ∈ {∅} → (∪ 𝐹 ∖ 𝑥) = ∅)
87		elssuni 4467	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐹 ∪ {∅}) → 𝑥 ⊆ ∪ (𝐹 ∪ {∅}))
88	87, 21	syl6sseqr 3652	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐹 ∪ {∅}) → 𝑥 ⊆ ∪ 𝐹)
89	88	adantl 482	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → 𝑥 ⊆ ∪ 𝐹)
90		ssdif0 3942	. . . . . . . . . . 11 ⊢ (∪ 𝐹 ⊆ 𝑥 ↔ (∪ 𝐹 ∖ 𝑥) = ∅)
91	90	biimpri 218	. . . . . . . . . 10 ⊢ ((∪ 𝐹 ∖ 𝑥) = ∅ → ∪ 𝐹 ⊆ 𝑥)
92		eqss 3618	. . . . . . . . . . 11 ⊢ (𝑥 = ∪ 𝐹 ↔ (𝑥 ⊆ ∪ 𝐹 ∧ ∪ 𝐹 ⊆ 𝑥))
93	92	simplbi2 655	. . . . . . . . . 10 ⊢ (𝑥 ⊆ ∪ 𝐹 → (∪ 𝐹 ⊆ 𝑥 → 𝑥 = ∪ 𝐹))
94	89, 91, 93	syl2im 40	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → ((∪ 𝐹 ∖ 𝑥) = ∅ → 𝑥 = ∪ 𝐹))
95	86, 94	syl5 34	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → ((∪ 𝐹 ∖ 𝑥) ∈ {∅} → 𝑥 = ∪ 𝐹))
96	85, 95	orim12d 883	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → (((∪ 𝐹 ∖ 𝑥) ∈ 𝐹 ∨ (∪ 𝐹 ∖ 𝑥) ∈ {∅}) → (𝑥 = ∅ ∨ 𝑥 = ∪ 𝐹)))
97	75, 96	syl5 34	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → (𝑥 ∈ (Clsd‘(𝐹 ∪ {∅})) → (𝑥 = ∅ ∨ 𝑥 = ∪ 𝐹)))
98	97	expimpd 629	. . . . 5 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → ((𝑥 ∈ (𝐹 ∪ {∅}) ∧ 𝑥 ∈ (Clsd‘(𝐹 ∪ {∅}))) → (𝑥 = ∅ ∨ 𝑥 = ∪ 𝐹)))
99		elin 3796	. . . . 5 ⊢ (𝑥 ∈ ((𝐹 ∪ {∅}) ∩ (Clsd‘(𝐹 ∪ {∅}))) ↔ (𝑥 ∈ (𝐹 ∪ {∅}) ∧ 𝑥 ∈ (Clsd‘(𝐹 ∪ {∅}))))
100		vex 3203	. . . . . 6 ⊢ 𝑥 ∈ V
101	100	elpr 4198	. . . . 5 ⊢ (𝑥 ∈ {∅, ∪ 𝐹} ↔ (𝑥 = ∅ ∨ 𝑥 = ∪ 𝐹))
102	98, 99, 101	3imtr4g 285	. . . 4 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → (𝑥 ∈ ((𝐹 ∪ {∅}) ∩ (Clsd‘(𝐹 ∪ {∅}))) → 𝑥 ∈ {∅, ∪ 𝐹}))
103	102	ssrdv 3609	. . 3 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → ((𝐹 ∪ {∅}) ∩ (Clsd‘(𝐹 ∪ {∅}))) ⊆ {∅, ∪ 𝐹})
104	21	isconn2 21217	. . 3 ⊢ ((𝐹 ∪ {∅}) ∈ Conn ↔ ((𝐹 ∪ {∅}) ∈ Top ∧ ((𝐹 ∪ {∅}) ∩ (Clsd‘(𝐹 ∪ {∅}))) ⊆ {∅, ∪ 𝐹}))
105	72, 103, 104	sylanbrc 698	. 2 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → (𝐹 ∪ {∅}) ∈ Conn)
106	4, 105	syl 17	1 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ {∅}) ∈ Conn)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {csn 4177  {cpr 4179  ∪ cuni 4436  ‘cfv 5888  fBascfbas 19734  Topctop 20698  Clsdccld 20820  Conncconn 21214  Filcfil 21649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-fbas 19743  df-top 20699  df-cld 20823  df-conn 21215  df-fil 21650

This theorem is referenced by:  ufildr  21735