pconnconn - Mathbox for Mario Carneiro

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Theorem pconnconn 33912

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 4311	. . . . . . . 8 ⊢ (𝑥 ≠ ∅ ↔ ∃𝑎 𝑎 ∈ 𝑥)
3		n0 4311	. . . . . . . 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 4875	. . . . . . . . . . 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 4875	. . . . . . . . . . 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 2731	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
17	16	pconncn 33905	. . . . . . . . . 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 24281	. . . . . . . . . . . . . . . . 17 ⊢ (0[,]1) = ∪ II
23		iiconn 24287	. . . . . . . . . . . . . . . . . 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 4118	. . . . . . . . . . . . . . . . . . . 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 2783	. . . . . . . . . . . . . . . . . . 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 4903	. . . . . . . . . . . . . . . . . . . . 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 4166	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∩ 𝑥) = (𝑥 ∩ 𝑦)
34	33, 19	eqtrid 2783	. . . . . . . . . . . . . . . . . . . 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 4455	. . . . . . . . . . . . . . . . . . . 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 33906	. . . . . . . . . . . . . . . . . . . 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 22421	. . . . . . . . . . . . . . . . . . 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 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)) → 𝑥 ∈ (Clsd‘𝐽))
43		0elunit 13396	. . . . . . . . . . . . . . . . . 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 2832	. . . . . . . . . . . . . . . . 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 22814	. . . . . . . . . . . . . . . 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 13397	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ (0[,]1)
51		ffvelcdm 7037	. . . . . . . . . . . . . . . 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 2833	. . . . . . . . . . . . . 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 4429	. . . . . . . . . . . . . 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 3025	. . . . . . . . . . . 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 2946	. . . . . . . . 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 3154	. . . . . . . 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 3193	. 2 ⊢ (𝐽 ∈ PConn → ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽))
68	16	toptopon 22303	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
69	38, 68	sylib 217	. . 3 ⊢ (𝐽 ∈ PConn → 𝐽 ∈ (TopOn‘∪ 𝐽))
70		dfconn2 22807	. . 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 2939 ∀wral 3060 ∃wrex 3069 ∖ cdif 3910 ∪ cun 3911 ∩ cin 3912 ⊆ wss 3913 ∅c0 4287 ∪ cuni 4870 ⟶wf 6497 ‘cfv 6501 (class class class)co 7362 0cc0 11060 1c1 11061 [,]cicc 13277 Topctop 22279 TopOnctopon 22296 Clsdccld 22404 Cn ccn 22612 Conncconn 22799 IIcii 24275 PConncpconn 33900

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 2702 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11116 ax-resscn 11117 ax-1cn 11118 ax-icn 11119 ax-addcl 11120 ax-addrcl 11121 ax-mulcl 11122 ax-mulrcl 11123 ax-mulcom 11124 ax-addass 11125 ax-mulass 11126 ax-distr 11127 ax-i2m1 11128 ax-1ne0 11129 ax-1rid 11130 ax-rnegex 11131 ax-rrecex 11132 ax-cnre 11133 ax-pre-lttri 11134 ax-pre-lttrn 11135 ax-pre-ltadd 11136 ax-pre-mulgt0 11137 ax-pre-sup 11138

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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3448 df-sbc 3743 df-csb 3859 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-map 8774 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fi 9356 df-sup 9387 df-inf 9388 df-pnf 11200 df-mnf 11201 df-xr 11202 df-ltxr 11203 df-le 11204 df-sub 11396 df-neg 11397 df-div 11822 df-nn 12163 df-2 12225 df-3 12226 df-n0 12423 df-z 12509 df-uz 12773 df-q 12883 df-rp 12925 df-xneg 13042 df-xadd 13043 df-xmul 13044 df-ioo 13278 df-ico 13280 df-icc 13281 df-seq 13917 df-exp 13978 df-cj 14996 df-re 14997 df-im 14998 df-sqrt 15132 df-abs 15133 df-rest 17318 df-topgen 17339 df-psmet 20825 df-xmet 20826 df-met 20827 df-bl 20828 df-mopn 20829 df-top 22280 df-topon 22297 df-bases 22333 df-cld 22407 df-cn 22615 df-conn 22800 df-ii 24277 df-pconn 33902

This theorem is referenced by: resconn 33927 iinllyconn 33935 cvmlift2lem10 33993 cvmlift3 34009

W3C validator