pconnconn - Mathbox for Mario Carneiro

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Theorem pconnconn 32478

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 1085	. . . 4 ⊢ ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) ↔ ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅) ∧ (𝑥 ∩ 𝑦) = ∅))
2		n0 4309	. . . . . . . 8 ⊢ (𝑥 ≠ ∅ ↔ ∃𝑎 𝑎 ∈ 𝑥)
3		n0 4309	. . . . . . . 8 ⊢ (𝑦 ≠ ∅ ↔ ∃𝑏 𝑏 ∈ 𝑦)
4	2, 3	anbi12i 628	. . . . . . 7 ⊢ ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅) ↔ (∃𝑎 𝑎 ∈ 𝑥 ∧ ∃𝑏 𝑏 ∈ 𝑦))
5		exdistrv 1952	. . . . . . 7 ⊢ (∃𝑎∃𝑏(𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ↔ (∃𝑎 𝑎 ∈ 𝑥 ∧ ∃𝑏 𝑏 ∈ 𝑦))
6	4, 5	bitr4i 280	. . . . . 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 4842	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝑥 ∧ 𝑥 ∈ 𝐽) → 𝑎 ∈ ∪ 𝐽)
11	8, 9, 10	syl2anc 586	. . . . . . . . . 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 4842	. . . . . . . . . . 11 ⊢ ((𝑏 ∈ 𝑦 ∧ 𝑦 ∈ 𝐽) → 𝑏 ∈ ∪ 𝐽)
15	12, 13, 14	syl2anc 586	. . . . . . . . . 10 ⊢ (((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑏 ∈ ∪ 𝐽)
16		eqid 2821	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
17	16	pconncn 32471	. . . . . . . . . 10 ⊢ ((𝐽 ∈ PConn ∧ 𝑎 ∈ ∪ 𝐽 ∧ 𝑏 ∈ ∪ 𝐽) → ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏))
18	7, 11, 15, 17	syl3anc 1367	. . . . . . . . 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 484	. . . . . . . . . . . . . . 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 23488	. . . . . . . . . . . . . . . . 17 ⊢ (0[,]1) = ∪ II
23		iiconn 23494	. . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . 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 4128	. . . . . . . . . . . . . . . . . . . 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	syl5eq 2868	. . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . 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 4867	. . . . . . . . . . . . . . . . . . . . 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 4177	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∩ 𝑥) = (𝑥 ∩ 𝑦)
34	33, 19	syl5eq 2868	. . . . . . . . . . . . . . . . . . . 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 4437	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ⊆ ∪ 𝐽 ∧ (𝑦 ∩ 𝑥) = ∅) → ((𝑦 ∪ 𝑥) = ∪ 𝐽 ↔ (∪ 𝐽 ∖ 𝑦) = 𝑥))
36	32, 34, 35	syl2anc 586	. . . . . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . . . . 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 32472	. . . . . . . . . . . . . . . . . . . 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 21640	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑦) ∈ (Clsd‘𝐽))
41	39, 30, 40	syl2anc 586	. . . . . . . . . . . . . . . . . 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 2914	. . . . . . . . . . . . . . . . 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 12854	. . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . 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 2913	. . . . . . . . . . . . . . . . 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 22033	. . . . . . . . . . . . . . . 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 12855	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ (0[,]1)
51		ffvelrn 6848	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(0[,]1)⟶𝑥 ∧ 1 ∈ (0[,]1)) → (𝑓‘1) ∈ 𝑥)
52	49, 50, 51	sylancl 588	. . . . . . . . . . . . . . 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 2914	. . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . 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 4413	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) → (𝑥 ∩ 𝑦) ≠ ∅)
56	53, 54, 55	syl2anc 586	. . . . . . . . . . . . 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 3101	. . . . . . . . . . . 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 459	. . . . . . . . . . 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 192	. . . . . . . . . 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 3023	. . . . . . . . 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 3291	. . . . . . . 8 ⊢ (((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽)
62	61	exp32 423	. . . . . . 7 ⊢ ((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) → ((𝑥 ∩ 𝑦) = ∅ → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽)))
63	62	exlimdvv 1931	. . . . . 6 ⊢ ((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → (∃𝑎∃𝑏(𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) → ((𝑥 ∩ 𝑦) = ∅ → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽)))
64	6, 63	syl5bi 244	. . . . 5 ⊢ ((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅) → ((𝑥 ∩ 𝑦) = ∅ → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽)))
65	64	impd 413	. . . 4 ⊢ ((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → (((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅) ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽))
66	1, 65	syl5bi 244	. . 3 ⊢ ((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽))
67	66	ralrimivva 3191	. 2 ⊢ (𝐽 ∈ PConn → ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽))
68	16	toptopon 21524	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
69	38, 68	sylib 220	. . 3 ⊢ (𝐽 ∈ PConn → 𝐽 ∈ (TopOn‘∪ 𝐽))
70		dfconn2 22026	. . 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 259	1 ⊢ (𝐽 ∈ PConn → 𝐽 ∈ Conn)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∃wex 1776 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 ∖ cdif 3932 ∪ cun 3933 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 ∪ cuni 4837 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 0cc0 10536 1c1 10537 [,]cicc 12740 Topctop 21500 TopOnctopon 21517 Clsdccld 21623 Cn ccn 21831 Conncconn 22018 IIcii 23482 PConncpconn 32466

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-oadd 8105 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fi 8874 df-sup 8905 df-inf 8906 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-z 11981 df-uz 12243 df-q 12348 df-rp 12389 df-xneg 12506 df-xadd 12507 df-xmul 12508 df-ioo 12741 df-ico 12743 df-icc 12744 df-seq 13369 df-exp 13429 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-rest 16695 df-topgen 16716 df-psmet 20536 df-xmet 20537 df-met 20538 df-bl 20539 df-mopn 20540 df-top 21501 df-topon 21518 df-bases 21553 df-cld 21626 df-cn 21834 df-conn 22019 df-ii 23484 df-pconn 32468

This theorem is referenced by: resconn 32493 iinllyconn 32501 cvmlift2lem10 32559 cvmlift3 32575

W3C validator