pconnconn - Mathbox for Mario Carneiro

	Mathbox for Mario Carneiro		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > pconnconn		Structured version Visualization version GIF version

Theorem pconnconn 34291

Description: A path-connected space is connected. (Contributed by Mario Carneiro, 11-Feb-2015.)

**Assertion**
Ref	Expression
pconnconn	⊢ (𝐽 ∈ PConn → 𝐽 ∈ Conn)

Proof of Theorem pconnconn Dummy variables 𝑎 𝑏 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-3an 1089	. . . 4 ⊢ ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) ↔ ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅) ∧ (𝑥 ∩ 𝑦) = ∅))
2		n0 4346	. . . . . . . 8 ⊢ (𝑥 ≠ ∅ ↔ ∃𝑎 𝑎 ∈ 𝑥)
3		n0 4346	. . . . . . . 8 ⊢ (𝑦 ≠ ∅ ↔ ∃𝑏 𝑏 ∈ 𝑦)
4	2, 3	anbi12i 627	. . . . . . 7 ⊢ ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅) ↔ (∃𝑎 𝑎 ∈ 𝑥 ∧ ∃𝑏 𝑏 ∈ 𝑦))
5		exdistrv 1959	. . . . . . 7 ⊢ (∃𝑎∃𝑏(𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ↔ (∃𝑎 𝑎 ∈ 𝑥 ∧ ∃𝑏 𝑏 ∈ 𝑦))
6	4, 5	bitr4i 277	. . . . . 6 ⊢ ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅) ↔ ∃𝑎∃𝑏(𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦))
7		simpll 765	. . . . . . . . . 10 ⊢ (((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝐽 ∈ PConn)
8		simprll 777	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑎 ∈ 𝑥)
9		simplrl 775	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑥 ∈ 𝐽)
10		elunii 4913	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝑥 ∧ 𝑥 ∈ 𝐽) → 𝑎 ∈ ∪ 𝐽)
11	8, 9, 10	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑎 ∈ ∪ 𝐽)
12		simprlr 778	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑏 ∈ 𝑦)
13		simplrr 776	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑦 ∈ 𝐽)
14		elunii 4913	. . . . . . . . . . 11 ⊢ ((𝑏 ∈ 𝑦 ∧ 𝑦 ∈ 𝐽) → 𝑏 ∈ ∪ 𝐽)
15	12, 13, 14	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑏 ∈ ∪ 𝐽)
16		eqid 2732	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
17	16	pconncn 34284	. . . . . . . . . 10 ⊢ ((𝐽 ∈ PConn ∧ 𝑎 ∈ ∪ 𝐽 ∧ 𝑏 ∈ ∪ 𝐽) → ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏))
18	7, 11, 15, 17	syl3anc 1371	. . . . . . . . 9 ⊢ (((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) → ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏))
19		simplrr 776	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) ∧ ((𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏)) ∧ (𝑥 ∪ 𝑦) = ∪ 𝐽)) → (𝑥 ∩ 𝑦) = ∅)
20		simplrr 776	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏)) ∧ (𝑥 ∪ 𝑦) = ∪ 𝐽) → (𝑓‘1) = 𝑏)
21	20	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) ∧ ((𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏)) ∧ (𝑥 ∪ 𝑦) = ∪ 𝐽)) → (𝑓‘1) = 𝑏)
22		iiuni 24404	. . . . . . . . . . . . . . . . 17 ⊢ (0[,]1) = ∪ II
23		iiconn 24410	. . . . . . . . . . . . . . . . . 18 ⊢ II ∈ Conn
24	23	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) ∧ ((𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏)) ∧ (𝑥 ∪ 𝑦) = ∪ 𝐽)) → II ∈ Conn)
25		simprll 777	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) ∧ ((𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏)) ∧ (𝑥 ∪ 𝑦) = ∪ 𝐽)) → 𝑓 ∈ (II Cn 𝐽))
26	9	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) ∧ ((𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏)) ∧ (𝑥 ∪ 𝑦) = ∪ 𝐽)) → 𝑥 ∈ 𝐽)
27		uncom 4153	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∪ 𝑥) = (𝑥 ∪ 𝑦)
28		simprr 771	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) ∧ ((𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏)) ∧ (𝑥 ∪ 𝑦) = ∪ 𝐽)) → (𝑥 ∪ 𝑦) = ∪ 𝐽)
29	27, 28	eqtrid 2784	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) ∧ ((𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏)) ∧ (𝑥 ∪ 𝑦) = ∪ 𝐽)) → (𝑦 ∪ 𝑥) = ∪ 𝐽)
30	13	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) ∧ ((𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏)) ∧ (𝑥 ∪ 𝑦) = ∪ 𝐽)) → 𝑦 ∈ 𝐽)
31		elssuni 4941	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ 𝐽 → 𝑦 ⊆ ∪ 𝐽)
32	30, 31	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) ∧ ((𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏)) ∧ (𝑥 ∪ 𝑦) = ∪ 𝐽)) → 𝑦 ⊆ ∪ 𝐽)
33		incom 4201	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∩ 𝑥) = (𝑥 ∩ 𝑦)
34	33, 19	eqtrid 2784	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) ∧ ((𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏)) ∧ (𝑥 ∪ 𝑦) = ∪ 𝐽)) → (𝑦 ∩ 𝑥) = ∅)
35		uneqdifeq 4492	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ⊆ ∪ 𝐽 ∧ (𝑦 ∩ 𝑥) = ∅) → ((𝑦 ∪ 𝑥) = ∪ 𝐽 ↔ (∪ 𝐽 ∖ 𝑦) = 𝑥))
36	32, 34, 35	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) ∧ ((𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏)) ∧ (𝑥 ∪ 𝑦) = ∪ 𝐽)) → ((𝑦 ∪ 𝑥) = ∪ 𝐽 ↔ (∪ 𝐽 ∖ 𝑦) = 𝑥))
37	29, 36	mpbid 231	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) ∧ ((𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏)) ∧ (𝑥 ∪ 𝑦) = ∪ 𝐽)) → (∪ 𝐽 ∖ 𝑦) = 𝑥)
38		pconntop 34285	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐽 ∈ PConn → 𝐽 ∈ Top)
39	38	ad3antrrr 728	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) ∧ ((𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏)) ∧ (𝑥 ∪ 𝑦) = ∪ 𝐽)) → 𝐽 ∈ Top)
40	16	opncld 22544	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑦) ∈ (Clsd‘𝐽))
41	39, 30, 40	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) ∧ ((𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏)) ∧ (𝑥 ∪ 𝑦) = ∪ 𝐽)) → (∪ 𝐽 ∖ 𝑦) ∈ (Clsd‘𝐽))
42	37, 41	eqeltrrd 2834	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) ∧ ((𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏)) ∧ (𝑥 ∪ 𝑦) = ∪ 𝐽)) → 𝑥 ∈ (Clsd‘𝐽))
43		0elunit 13448	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ (0[,]1)
44	43	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) ∧ ((𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏)) ∧ (𝑥 ∪ 𝑦) = ∪ 𝐽)) → 0 ∈ (0[,]1))
45		simplrl 775	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏)) ∧ (𝑥 ∪ 𝑦) = ∪ 𝐽) → (𝑓‘0) = 𝑎)
46	45	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) ∧ ((𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏)) ∧ (𝑥 ∪ 𝑦) = ∪ 𝐽)) → (𝑓‘0) = 𝑎)
47	8	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) ∧ ((𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏)) ∧ (𝑥 ∪ 𝑦) = ∪ 𝐽)) → 𝑎 ∈ 𝑥)
48	46, 47	eqeltrd 2833	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) ∧ ((𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏)) ∧ (𝑥 ∪ 𝑦) = ∪ 𝐽)) → (𝑓‘0) ∈ 𝑥)
49	22, 24, 25, 26, 42, 44, 48	conncn 22937	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) ∧ ((𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏)) ∧ (𝑥 ∪ 𝑦) = ∪ 𝐽)) → 𝑓:(0[,]1)⟶𝑥)
50		1elunit 13449	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ (0[,]1)
51		ffvelcdm 7083	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(0[,]1)⟶𝑥 ∧ 1 ∈ (0[,]1)) → (𝑓‘1) ∈ 𝑥)
52	49, 50, 51	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) ∧ ((𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏)) ∧ (𝑥 ∪ 𝑦) = ∪ 𝐽)) → (𝑓‘1) ∈ 𝑥)
53	21, 52	eqeltrrd 2834	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) ∧ ((𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏)) ∧ (𝑥 ∪ 𝑦) = ∪ 𝐽)) → 𝑏 ∈ 𝑥)
54	12	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) ∧ ((𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏)) ∧ (𝑥 ∪ 𝑦) = ∪ 𝐽)) → 𝑏 ∈ 𝑦)
55		inelcm 4464	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) → (𝑥 ∩ 𝑦) ≠ ∅)
56	53, 54, 55	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) ∧ ((𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏)) ∧ (𝑥 ∪ 𝑦) = ∪ 𝐽)) → (𝑥 ∩ 𝑦) ≠ ∅)
57	19, 56	pm2.21ddne 3026	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) ∧ ((𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏)) ∧ (𝑥 ∪ 𝑦) = ∪ 𝐽)) → ¬ (𝑥 ∪ 𝑦) = ∪ 𝐽)
58	57	expr 457	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) ∧ (𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏))) → ((𝑥 ∪ 𝑦) = ∪ 𝐽 → ¬ (𝑥 ∪ 𝑦) = ∪ 𝐽))
59	58	pm2.01d 189	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) ∧ (𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏))) → ¬ (𝑥 ∪ 𝑦) = ∪ 𝐽)
60	59	neqned 2947	. . . . . . . . 9 ⊢ ((((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) ∧ (𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏))) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽)
61	18, 60	rexlimddv 3161	. . . . . . . 8 ⊢ (((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽)
62	61	exp32 421	. . . . . . 7 ⊢ ((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) → ((𝑥 ∩ 𝑦) = ∅ → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽)))
63	62	exlimdvv 1937	. . . . . 6 ⊢ ((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → (∃𝑎∃𝑏(𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) → ((𝑥 ∩ 𝑦) = ∅ → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽)))
64	6, 63	biimtrid 241	. . . . 5 ⊢ ((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅) → ((𝑥 ∩ 𝑦) = ∅ → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽)))
65	64	impd 411	. . . 4 ⊢ ((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → (((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅) ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽))
66	1, 65	biimtrid 241	. . 3 ⊢ ((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽))
67	66	ralrimivva 3200	. 2 ⊢ (𝐽 ∈ PConn → ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽))
68	16	toptopon 22426	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
69	38, 68	sylib 217	. . 3 ⊢ (𝐽 ∈ PConn → 𝐽 ∈ (TopOn‘∪ 𝐽))
70		dfconn2 22930	. . 3 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (𝐽 ∈ Conn ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽)))
71	69, 70	syl 17	. 2 ⊢ (𝐽 ∈ PConn → (𝐽 ∈ Conn ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽)))
72	67, 71	mpbird 256	1 ⊢ (𝐽 ∈ PConn → 𝐽 ∈ Conn)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ∖ cdif 3945 ∪ cun 3946 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 ∪ cuni 4908 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 0cc0 11112 1c1 11113 [,]cicc 13329 Topctop 22402 TopOnctopon 22419 Clsdccld 22527 Cn ccn 22735 Conncconn 22922 IIcii 24398 PConncpconn 34279

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fi 9408 df-sup 9439 df-inf 9440 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-n0 12475 df-z 12561 df-uz 12825 df-q 12935 df-rp 12977 df-xneg 13094 df-xadd 13095 df-xmul 13096 df-ioo 13330 df-ico 13332 df-icc 13333 df-seq 13969 df-exp 14030 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-rest 17370 df-topgen 17391 df-psmet 20942 df-xmet 20943 df-met 20944 df-bl 20945 df-mopn 20946 df-top 22403 df-topon 22420 df-bases 22456 df-cld 22530 df-cn 22738 df-conn 22923 df-ii 24400 df-pconn 34281

This theorem is referenced by: resconn 34306 iinllyconn 34314 cvmlift2lem10 34372 cvmlift3 34388

W3C validator