pconnconn - Mathbox for Mario Carneiro

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Theorem pconnconn 33093

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 1087	. . . 4 ⊢ ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) ↔ ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅) ∧ (𝑥 ∩ 𝑦) = ∅))
2		n0 4277	. . . . . . . 8 ⊢ (𝑥 ≠ ∅ ↔ ∃𝑎 𝑎 ∈ 𝑥)
3		n0 4277	. . . . . . . 8 ⊢ (𝑦 ≠ ∅ ↔ ∃𝑏 𝑏 ∈ 𝑦)
4	2, 3	anbi12i 626	. . . . . . 7 ⊢ ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅) ↔ (∃𝑎 𝑎 ∈ 𝑥 ∧ ∃𝑏 𝑏 ∈ 𝑦))
5		exdistrv 1960	. . . . . . 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 763	. . . . . . . . . 10 ⊢ (((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝐽 ∈ PConn)
8		simprll 775	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑎 ∈ 𝑥)
9		simplrl 773	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑥 ∈ 𝐽)
10		elunii 4841	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝑥 ∧ 𝑥 ∈ 𝐽) → 𝑎 ∈ ∪ 𝐽)
11	8, 9, 10	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑎 ∈ ∪ 𝐽)
12		simprlr 776	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑏 ∈ 𝑦)
13		simplrr 774	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑦 ∈ 𝐽)
14		elunii 4841	. . . . . . . . . . 11 ⊢ ((𝑏 ∈ 𝑦 ∧ 𝑦 ∈ 𝐽) → 𝑏 ∈ ∪ 𝐽)
15	12, 13, 14	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑏 ∈ ∪ 𝐽)
16		eqid 2738	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
17	16	pconncn 33086	. . . . . . . . . 10 ⊢ ((𝐽 ∈ PConn ∧ 𝑎 ∈ ∪ 𝐽 ∧ 𝑏 ∈ ∪ 𝐽) → ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏))
18	7, 11, 15, 17	syl3anc 1369	. . . . . . . . 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 774	. . . . . . . . . . . . 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 774	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏)) ∧ (𝑥 ∪ 𝑦) = ∪ 𝐽) → (𝑓‘1) = 𝑏)
21	20	adantl 481	. . . . . . . . . . . . . . 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 23950	. . . . . . . . . . . . . . . . 17 ⊢ (0[,]1) = ∪ II
23		iiconn 23956	. . . . . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 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 4083	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∪ 𝑥) = (𝑥 ∪ 𝑦)
28		simprr 769	. . . . . . . . . . . . . . . . . . . 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 2791	. . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . 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 4868	. . . . . . . . . . . . . . . . . . . . 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 4131	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∩ 𝑥) = (𝑥 ∩ 𝑦)
34	33, 19	syl5eq 2791	. . . . . . . . . . . . . . . . . . . 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 4420	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ⊆ ∪ 𝐽 ∧ (𝑦 ∩ 𝑥) = ∅) → ((𝑦 ∪ 𝑥) = ∪ 𝐽 ↔ (∪ 𝐽 ∖ 𝑦) = 𝑥))
36	32, 34, 35	syl2anc 583	. . . . . . . . . . . . . . . . . . 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 33087	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐽 ∈ PConn → 𝐽 ∈ Top)
39	38	ad3antrrr 726	. . . . . . . . . . . . . . . . . . 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 22092	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑦) ∈ (Clsd‘𝐽))
41	39, 30, 40	syl2anc 583	. . . . . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . . . . 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 13130	. . . . . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓 ∈ (II Cn 𝐽) ∧ ((𝑓‘0) = 𝑎 ∧ (𝑓‘1) = 𝑏)) ∧ (𝑥 ∪ 𝑦) = ∪ 𝐽) → (𝑓‘0) = 𝑎)
46	45	adantl 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)) → (𝑓‘0) = 𝑎)
47	8	adantr 480	. . . . . . . . . . . . . . . . . 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 2839	. . . . . . . . . . . . . . . . 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 22485	. . . . . . . . . . . . . . . 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 13131	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ (0[,]1)
51		ffvelrn 6941	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(0[,]1)⟶𝑥 ∧ 1 ∈ (0[,]1)) → (𝑓‘1) ∈ 𝑥)
52	49, 50, 51	sylancl 585	. . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 4395	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) → (𝑥 ∩ 𝑦) ≠ ∅)
56	53, 54, 55	syl2anc 583	. . . . . . . . . . . . 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 3028	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 2949	. . . . . . . . 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 3219	. . . . . . . 8 ⊢ (((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽)
62	61	exp32 420	. . . . . . 7 ⊢ ((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) → ((𝑥 ∩ 𝑦) = ∅ → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽)))
63	62	exlimdvv 1938	. . . . . 6 ⊢ ((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → (∃𝑎∃𝑏(𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) → ((𝑥 ∩ 𝑦) = ∅ → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽)))
64	6, 63	syl5bi 241	. . . . 5 ⊢ ((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅) → ((𝑥 ∩ 𝑦) = ∅ → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽)))
65	64	impd 410	. . . 4 ⊢ ((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → (((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅) ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽))
66	1, 65	syl5bi 241	. . 3 ⊢ ((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽))
67	66	ralrimivva 3114	. 2 ⊢ (𝐽 ∈ PConn → ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽))
68	16	toptopon 21974	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
69	38, 68	sylib 217	. . 3 ⊢ (𝐽 ∈ PConn → 𝐽 ∈ (TopOn‘∪ 𝐽))
70		dfconn2 22478	. . 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 395 ∧ w3a 1085 = wceq 1539 ∃wex 1783 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 ∖ cdif 3880 ∪ cun 3881 ∩ cin 3882 ⊆ wss 3883 ∅c0 4253 ∪ cuni 4836 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 0cc0 10802 1c1 10803 [,]cicc 13011 Topctop 21950 TopOnctopon 21967 Clsdccld 22075 Cn ccn 22283 Conncconn 22470 IIcii 23944 PConncpconn 33081

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fi 9100 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ico 13014 df-icc 13015 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-rest 17050 df-topgen 17071 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-top 21951 df-topon 21968 df-bases 22004 df-cld 22078 df-cn 22286 df-conn 22471 df-ii 23946 df-pconn 33083

This theorem is referenced by: resconn 33108 iinllyconn 33116 cvmlift2lem10 33174 cvmlift3 33190

W3C validator