Theorem dfconn2 23606

Description: An alternate definition of connectedness. (Contributed by Jeff Hankins, 9-Jul-2009.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)

Assertion

Ref	Expression
dfconn2	⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Conn ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ 𝑋)))

Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝑋,𝑦

Proof of Theorem dfconn2

Step	Hyp	Ref	Expression
1		eqid 2765	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
2		simpll 779	. . . . . 6 ⊢ (((𝐽 ∈ Conn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝐽 ∈ Conn)
3		simplrl 789	. . . . . 6 ⊢ (((𝐽 ∈ Conn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑥 ∈ 𝐽)
4		simpr1 1213	. . . . . 6 ⊢ (((𝐽 ∈ Conn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑥 ≠ ∅)
5		simplrr 790	. . . . . 6 ⊢ (((𝐽 ∈ Conn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑦 ∈ 𝐽)
6		simpr2 1214	. . . . . 6 ⊢ (((𝐽 ∈ Conn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑦 ≠ ∅)
7		simpr3 1215	. . . . . 6 ⊢ (((𝐽 ∈ Conn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅)) → (𝑥 ∩ 𝑦) = ∅)
8	1, 2, 3, 4, 5, 6, 7	conndisj 23603	. . . . 5 ⊢ (((𝐽 ∈ Conn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅)) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽)
9	8	ex 418	. . . 4 ⊢ ((𝐽 ∈ Conn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽))
10	9	ralrimivva 3210	. . 3 ⊢ (𝐽 ∈ Conn → ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽))
11		topontop 23100	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
12	1	cldopn 23218	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ 𝑥) ∈ 𝐽)
13	12	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ 𝑥) ∈ 𝐽)
14		df-3an 1105	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) ↔ ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅) ∧ (𝑥 ∩ 𝑦) = ∅))
15		ineq2 4167	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (∪ 𝐽 ∖ 𝑥) → (𝑥 ∩ 𝑦) = (𝑥 ∩ (∪ 𝐽 ∖ 𝑥)))
16		disjdif 4433	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∩ (∪ 𝐽 ∖ 𝑥)) = ∅
17	15, 16	eqtrdi 2816	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (∪ 𝐽 ∖ 𝑥) → (𝑥 ∩ 𝑦) = ∅)
18	17	biantrud 541	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (∪ 𝐽 ∖ 𝑥) → ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅) ↔ ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅) ∧ (𝑥 ∩ 𝑦) = ∅)))
19		neeq1 3022	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (∪ 𝐽 ∖ 𝑥) → (𝑦 ≠ ∅ ↔ (∪ 𝐽 ∖ 𝑥) ≠ ∅))
20	19	anbi2d 642	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (∪ 𝐽 ∖ 𝑥) → ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅) ↔ (𝑥 ≠ ∅ ∧ (∪ 𝐽 ∖ 𝑥) ≠ ∅)))
21	18, 20	bitr3d 284	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (∪ 𝐽 ∖ 𝑥) → (((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅) ∧ (𝑥 ∩ 𝑦) = ∅) ↔ (𝑥 ≠ ∅ ∧ (∪ 𝐽 ∖ 𝑥) ≠ ∅)))
22	14, 21	bitrid 286	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (∪ 𝐽 ∖ 𝑥) → ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) ↔ (𝑥 ≠ ∅ ∧ (∪ 𝐽 ∖ 𝑥) ≠ ∅)))
23		uneq2 4116	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (∪ 𝐽 ∖ 𝑥) → (𝑥 ∪ 𝑦) = (𝑥 ∪ (∪ 𝐽 ∖ 𝑥)))
24		undif2 4438	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∪ (∪ 𝐽 ∖ 𝑥)) = (𝑥 ∪ ∪ 𝐽)
25	23, 24	eqtrdi 2816	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (∪ 𝐽 ∖ 𝑥) → (𝑥 ∪ 𝑦) = (𝑥 ∪ ∪ 𝐽))
26	25	neeq1d 3019	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (∪ 𝐽 ∖ 𝑥) → ((𝑥 ∪ 𝑦) ≠ ∪ 𝐽 ↔ (𝑥 ∪ ∪ 𝐽) ≠ ∪ 𝐽))
27	22, 26	imbi12d 347	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (∪ 𝐽 ∖ 𝑥) → (((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽) ↔ ((𝑥 ≠ ∅ ∧ (∪ 𝐽 ∖ 𝑥) ≠ ∅) → (𝑥 ∪ ∪ 𝐽) ≠ ∪ 𝐽)))
28	27	rspcv 3579	. . . . . . . . . . . . 13 ⊢ ((∪ 𝐽 ∖ 𝑥) ∈ 𝐽 → (∀𝑦 ∈ 𝐽 ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽) → ((𝑥 ≠ ∅ ∧ (∪ 𝐽 ∖ 𝑥) ≠ ∅) → (𝑥 ∪ ∪ 𝐽) ≠ ∪ 𝐽)))
29	13, 28	syl 18	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (∀𝑦 ∈ 𝐽 ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽) → ((𝑥 ≠ ∅ ∧ (∪ 𝐽 ∖ 𝑥) ≠ ∅) → (𝑥 ∪ ∪ 𝐽) ≠ ∪ 𝐽)))
30	1	cldss 23216	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ⊆ ∪ 𝐽)
31	30	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 ⊆ ∪ 𝐽)
32		ssequn1 4139	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ⊆ ∪ 𝐽 ↔ (𝑥 ∪ ∪ 𝐽) = ∪ 𝐽)
33	31, 32	sylib 221	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑥 ∪ ∪ 𝐽) = ∪ 𝐽)
34		ssdif0 4321	. . . . . . . . . . . . . . . 16 ⊢ (∪ 𝐽 ⊆ 𝑥 ↔ (∪ 𝐽 ∖ 𝑥) = ∅)
35		idd 25	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ⊆ 𝑥 → ∪ 𝐽 ⊆ 𝑥))
36	35, 31	jctild 535	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ⊆ 𝑥 → (𝑥 ⊆ ∪ 𝐽 ∧ ∪ 𝐽 ⊆ 𝑥)))
37		eqss 3953	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ∪ 𝐽 ↔ (𝑥 ⊆ ∪ 𝐽 ∧ ∪ 𝐽 ⊆ 𝑥))
38	36, 37	imbitrrdi 255	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ⊆ 𝑥 → 𝑥 = ∪ 𝐽))
39	34, 38	biimtrrid 246	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → ((∪ 𝐽 ∖ 𝑥) = ∅ → 𝑥 = ∪ 𝐽))
40	33, 39	embantd 60	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (((𝑥 ∪ ∪ 𝐽) = ∪ 𝐽 → (∪ 𝐽 ∖ 𝑥) = ∅) → 𝑥 = ∪ 𝐽))
41	40	orim2d 982	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → ((𝑥 = ∅ ∨ ((𝑥 ∪ ∪ 𝐽) = ∪ 𝐽 → (∪ 𝐽 ∖ 𝑥) = ∅)) → (𝑥 = ∅ ∨ 𝑥 = ∪ 𝐽)))
42		impexp 456	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ≠ ∅ ∧ (∪ 𝐽 ∖ 𝑥) ≠ ∅) → (𝑥 ∪ ∪ 𝐽) ≠ ∪ 𝐽) ↔ (𝑥 ≠ ∅ → ((∪ 𝐽 ∖ 𝑥) ≠ ∅ → (𝑥 ∪ ∪ 𝐽) ≠ ∪ 𝐽)))
43		df-ne 2961	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
44		id 23	. . . . . . . . . . . . . . . . . 18 ⊢ (((∪ 𝐽 ∖ 𝑥) ≠ ∅ → (𝑥 ∪ ∪ 𝐽) ≠ ∪ 𝐽) → ((∪ 𝐽 ∖ 𝑥) ≠ ∅ → (𝑥 ∪ ∪ 𝐽) ≠ ∪ 𝐽))
45	44	necon4d 2984	. . . . . . . . . . . . . . . . 17 ⊢ (((∪ 𝐽 ∖ 𝑥) ≠ ∅ → (𝑥 ∪ ∪ 𝐽) ≠ ∪ 𝐽) → ((𝑥 ∪ ∪ 𝐽) = ∪ 𝐽 → (∪ 𝐽 ∖ 𝑥) = ∅))
46		id 23	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∪ ∪ 𝐽) = ∪ 𝐽 → (∪ 𝐽 ∖ 𝑥) = ∅) → ((𝑥 ∪ ∪ 𝐽) = ∪ 𝐽 → (∪ 𝐽 ∖ 𝑥) = ∅))
47	46	necon3d 2981	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∪ ∪ 𝐽) = ∪ 𝐽 → (∪ 𝐽 ∖ 𝑥) = ∅) → ((∪ 𝐽 ∖ 𝑥) ≠ ∅ → (𝑥 ∪ ∪ 𝐽) ≠ ∪ 𝐽))
48	45, 47	impbii 212	. . . . . . . . . . . . . . . 16 ⊢ (((∪ 𝐽 ∖ 𝑥) ≠ ∅ → (𝑥 ∪ ∪ 𝐽) ≠ ∪ 𝐽) ↔ ((𝑥 ∪ ∪ 𝐽) = ∪ 𝐽 → (∪ 𝐽 ∖ 𝑥) = ∅))
49	43, 48	imbi12i 353	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ≠ ∅ → ((∪ 𝐽 ∖ 𝑥) ≠ ∅ → (𝑥 ∪ ∪ 𝐽) ≠ ∪ 𝐽)) ↔ (¬ 𝑥 = ∅ → ((𝑥 ∪ ∪ 𝐽) = ∪ 𝐽 → (∪ 𝐽 ∖ 𝑥) = ∅)))
50		pm4.64 863	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝑥 = ∅ → ((𝑥 ∪ ∪ 𝐽) = ∪ 𝐽 → (∪ 𝐽 ∖ 𝑥) = ∅)) ↔ (𝑥 = ∅ ∨ ((𝑥 ∪ ∪ 𝐽) = ∪ 𝐽 → (∪ 𝐽 ∖ 𝑥) = ∅)))
51	49, 50	bitri 278	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ≠ ∅ → ((∪ 𝐽 ∖ 𝑥) ≠ ∅ → (𝑥 ∪ ∪ 𝐽) ≠ ∪ 𝐽)) ↔ (𝑥 = ∅ ∨ ((𝑥 ∪ ∪ 𝐽) = ∪ 𝐽 → (∪ 𝐽 ∖ 𝑥) = ∅)))
52	42, 51	bitri 278	. . . . . . . . . . . . 13 ⊢ (((𝑥 ≠ ∅ ∧ (∪ 𝐽 ∖ 𝑥) ≠ ∅) → (𝑥 ∪ ∪ 𝐽) ≠ ∪ 𝐽) ↔ (𝑥 = ∅ ∨ ((𝑥 ∪ ∪ 𝐽) = ∪ 𝐽 → (∪ 𝐽 ∖ 𝑥) = ∅)))
53		vex 3461	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
54	53	elpr 4616	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ {∅, ∪ 𝐽} ↔ (𝑥 = ∅ ∨ 𝑥 = ∪ 𝐽))
55	41, 52, 54	3imtr4g 299	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (((𝑥 ≠ ∅ ∧ (∪ 𝐽 ∖ 𝑥) ≠ ∅) → (𝑥 ∪ ∪ 𝐽) ≠ ∪ 𝐽) → 𝑥 ∈ {∅, ∪ 𝐽}))
56	29, 55	syld 48	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (∀𝑦 ∈ 𝐽 ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽) → 𝑥 ∈ {∅, ∪ 𝐽}))
57	56	ex 418	. . . . . . . . . 10 ⊢ (𝐽 ∈ Top → (𝑥 ∈ (Clsd‘𝐽) → (∀𝑦 ∈ 𝐽 ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽) → 𝑥 ∈ {∅, ∪ 𝐽})))
58	57	com23 87	. . . . . . . . 9 ⊢ (𝐽 ∈ Top → (∀𝑦 ∈ 𝐽 ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽) → (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ∈ {∅, ∪ 𝐽})))
59	58	imim2d 58	. . . . . . . 8 ⊢ (𝐽 ∈ Top → ((𝑥 ∈ 𝐽 → ∀𝑦 ∈ 𝐽 ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽)) → (𝑥 ∈ 𝐽 → (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ∈ {∅, ∪ 𝐽}))))
60		elin 3922	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐽 ∩ (Clsd‘𝐽)) ↔ (𝑥 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽)))
61	60	imbi1i 352	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝐽 ∩ (Clsd‘𝐽)) → 𝑥 ∈ {∅, ∪ 𝐽}) ↔ ((𝑥 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 ∈ {∅, ∪ 𝐽}))
62		impexp 456	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 ∈ {∅, ∪ 𝐽}) ↔ (𝑥 ∈ 𝐽 → (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ∈ {∅, ∪ 𝐽})))
63	61, 62	bitri 278	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐽 ∩ (Clsd‘𝐽)) → 𝑥 ∈ {∅, ∪ 𝐽}) ↔ (𝑥 ∈ 𝐽 → (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ∈ {∅, ∪ 𝐽})))
64	59, 63	imbitrrdi 255	. . . . . . 7 ⊢ (𝐽 ∈ Top → ((𝑥 ∈ 𝐽 → ∀𝑦 ∈ 𝐽 ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽)) → (𝑥 ∈ (𝐽 ∩ (Clsd‘𝐽)) → 𝑥 ∈ {∅, ∪ 𝐽})))
65	64	alimdv 1949	. . . . . 6 ⊢ (𝐽 ∈ Top → (∀𝑥(𝑥 ∈ 𝐽 → ∀𝑦 ∈ 𝐽 ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽)) → ∀𝑥(𝑥 ∈ (𝐽 ∩ (Clsd‘𝐽)) → 𝑥 ∈ {∅, ∪ 𝐽})))
66		df-ral 3082	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽) ↔ ∀𝑥(𝑥 ∈ 𝐽 → ∀𝑦 ∈ 𝐽 ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽)))
67		df-ss 3923	. . . . . 6 ⊢ ((𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, ∪ 𝐽} ↔ ∀𝑥(𝑥 ∈ (𝐽 ∩ (Clsd‘𝐽)) → 𝑥 ∈ {∅, ∪ 𝐽}))
68	65, 66, 67	3imtr4g 299	. . . . 5 ⊢ (𝐽 ∈ Top → (∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽) → (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, ∪ 𝐽}))
69	1	isconn2 23601	. . . . . 6 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, ∪ 𝐽}))
70	69	baib 545	. . . . 5 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Conn ↔ (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, ∪ 𝐽}))
71	68, 70	sylibrd 262	. . . 4 ⊢ (𝐽 ∈ Top → (∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽) → 𝐽 ∈ Conn))
72	11, 71	syl 18	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽) → 𝐽 ∈ Conn))
73	10, 72	impbid2 229	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Conn ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽)))
74		toponuni 23101	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
75	74	neeq2d 3020	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ((𝑥 ∪ 𝑦) ≠ 𝑋 ↔ (𝑥 ∪ 𝑦) ≠ ∪ 𝐽))
76	75	imbi2d 343	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ 𝑋) ↔ ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽)))
77	76	2ralbidv 3231	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ 𝑋) ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽)))
78	73, 77	bitr4d 285	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Conn ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ 𝑋)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081   ∖ cdif 3903   ∪ cun 3904   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286  {cpr 4593  ∪ cuni 4874  ‘cfv 6540  Topctop 23080  TopOnctopon 23097  Clsdccld 23203  Conncconn 23598

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fn 6543  df-fv 6548  df-top 23081  df-topon 23098  df-cld 23206  df-conn 23599

This theorem is used by:  connsuba  23607  pconnconn  35736