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 35199

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 4376	. . . . . . . 8 ⊢ (𝑥 ≠ ∅ ↔ ∃𝑎 𝑎 ∈ 𝑥)
3		n0 4376	. . . . . . . 8 ⊢ (𝑦 ≠ ∅ ↔ ∃𝑏 𝑏 ∈ 𝑦)
4	2, 3	anbi12i 627	. . . . . . 7 ⊢ ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅) ↔ (∃𝑎 𝑎 ∈ 𝑥 ∧ ∃𝑏 𝑏 ∈ 𝑦))
5		exdistrv 1955	. . . . . . 7 ⊢ (∃𝑎∃𝑏(𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ↔ (∃𝑎 𝑎 ∈ 𝑥 ∧ ∃𝑏 𝑏 ∈ 𝑦))
6	4, 5	bitr4i 278	. . . . . 6 ⊢ ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅) ↔ ∃𝑎∃𝑏(𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦))
7		simpll 766	. . . . . . . . . 10 ⊢ (((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝐽 ∈ PConn)
8		simprll 778	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑎 ∈ 𝑥)
9		simplrl 776	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑥 ∈ 𝐽)
10		elunii 4936	. . . . . . . . . . 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 779	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑏 ∈ 𝑦)
13		simplrr 777	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑦 ∈ 𝐽)
14		elunii 4936	. . . . . . . . . . 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 2740	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
17	16	pconncn 35192	. . . . . . . . . 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 777	. . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . . 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 24926	. . . . . . . . . . . . . . . . 17 ⊢ (0[,]1) = ∪ II
23		iiconn 24932	. . . . . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . . . 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 4181	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∪ 𝑥) = (𝑥 ∪ 𝑦)
28		simprr 772	. . . . . . . . . . . . . . . . . . . 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 2792	. . . . . . . . . . . . . . . . . . 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 4961	. . . . . . . . . . . . . . . . . . . . 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 4230	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∩ 𝑥) = (𝑥 ∩ 𝑦)
34	33, 19	eqtrid 2792	. . . . . . . . . . . . . . . . . . . 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 4516	. . . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . 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 35193	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐽 ∈ PConn → 𝐽 ∈ Top)
39	38	ad3antrrr 729	. . . . . . . . . . . . . . . . . . 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 23062	. . . . . . . . . . . . . . . . . . 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 2845	. . . . . . . . . . . . . . . . 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 13529	. . . . . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . . . . 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 2844	. . . . . . . . . . . . . . . . 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 23455	. . . . . . . . . . . . . . . 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 13530	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ (0[,]1)
51		ffvelcdm 7115	. . . . . . . . . . . . . . . 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 2845	. . . . . . . . . . . . . 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 4488	. . . . . . . . . . . . . 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 3032	. . . . . . . . . . . 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 190	. . . . . . . . . 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 2953	. . . . . . . . 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 3167	. . . . . . . 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 1933	. . . . . 6 ⊢ ((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → (∃𝑎∃𝑏(𝑎 ∈ 𝑥 ∧ 𝑏 ∈ 𝑦) → ((𝑥 ∩ 𝑦) = ∅ → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽)))
64	6, 63	biimtrid 242	. . . . 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	biimtrid 242	. . 3 ⊢ ((𝐽 ∈ PConn ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽))
67	66	ralrimivva 3208	. 2 ⊢ (𝐽 ∈ PConn → ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 ((𝑥 ≠ ∅ ∧ 𝑦 ≠ ∅ ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∪ 𝑦) ≠ ∪ 𝐽))
68	16	toptopon 22944	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
69	38, 68	sylib 218	. . 3 ⊢ (𝐽 ∈ PConn → 𝐽 ∈ (TopOn‘∪ 𝐽))
70		dfconn2 23448	. . 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 257	1 ⊢ (𝐽 ∈ PConn → 𝐽 ∈ Conn)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∃wex 1777 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 ∖ cdif 3973 ∪ cun 3974 ∩ cin 3975 ⊆ wss 3976 ∅c0 4352 ∪ cuni 4931 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 0cc0 11184 1c1 11185 [,]cicc 13410 Topctop 22920 TopOnctopon 22937 Clsdccld 23045 Cn ccn 23253 Conncconn 23440 IIcii 24920 PConncpconn 35187

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fi 9480 df-sup 9511 df-inf 9512 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-z 12640 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ico 13413 df-icc 13414 df-seq 14053 df-exp 14113 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-rest 17482 df-topgen 17503 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-top 22921 df-topon 22938 df-bases 22974 df-cld 23048 df-cn 23256 df-conn 23441 df-ii 24922 df-pconn 35189

This theorem is referenced by: resconn 35214 iinllyconn 35222 cvmlift2lem10 35280 cvmlift3 35296

W3C validator