Theorem conndisj 23425

Description: If a topology is connected, its underlying set can't be partitioned into two nonempty non-overlapping open sets. (Contributed by FL, 16-Nov-2008.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)

Hypotheses

Ref	Expression
isconn.1	⊢ 𝑋 = ∪ 𝐽
connclo.1	⊢ (𝜑 → 𝐽 ∈ Conn)
connclo.2	⊢ (𝜑 → 𝐴 ∈ 𝐽)
connclo.3	⊢ (𝜑 → 𝐴 ≠ ∅)
conndisj.4	⊢ (𝜑 → 𝐵 ∈ 𝐽)
conndisj.5	⊢ (𝜑 → 𝐵 ≠ ∅)
conndisj.6	⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)

Assertion

Ref	Expression
conndisj	⊢ (𝜑 → (𝐴 ∪ 𝐵) ≠ 𝑋)

Proof of Theorem conndisj

Step	Hyp	Ref	Expression
1		connclo.3	. 2 ⊢ (𝜑 → 𝐴 ≠ ∅)
2		connclo.2	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝐽)
3		elssuni 4936	. . . . . . 7 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽)
4	2, 3	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽)
5		isconn.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
6	4, 5	sseqtrrdi 4024	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
7		conndisj.6	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
8		uneqdifeq 4492	. . . . 5 ⊢ ((𝐴 ⊆ 𝑋 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∪ 𝐵) = 𝑋 ↔ (𝑋 ∖ 𝐴) = 𝐵))
9	6, 7, 8	syl2anc 584	. . . 4 ⊢ (𝜑 → ((𝐴 ∪ 𝐵) = 𝑋 ↔ (𝑋 ∖ 𝐴) = 𝐵))
10		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ 𝐴) = 𝐵)
11	10	difeq2d 4125	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ (𝑋 ∖ 𝐴)) = (𝑋 ∖ 𝐵))
12		dfss4 4268	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴)
13	6, 12	sylib 218	. . . . . . 7 ⊢ (𝜑 → (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴)
14	13	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴)
15		connclo.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ Conn)
16	15	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐽 ∈ Conn)
17		conndisj.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ 𝐽)
18	17	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐵 ∈ 𝐽)
19		conndisj.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ≠ ∅)
20	19	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐵 ≠ ∅)
21	5	isconn 23422	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
22	21	simplbi 497	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ Conn → 𝐽 ∈ Top)
23	15, 22	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ Top)
24	5	opncld 23042	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → (𝑋 ∖ 𝐴) ∈ (Clsd‘𝐽))
25	23, 2, 24	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ∖ 𝐴) ∈ (Clsd‘𝐽))
26	25	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ 𝐴) ∈ (Clsd‘𝐽))
27	10, 26	eqeltrrd 2841	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐵 ∈ (Clsd‘𝐽))
28	5, 16, 18, 20, 27	connclo 23424	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐵 = 𝑋)
29	28	difeq2d 4125	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ 𝐵) = (𝑋 ∖ 𝑋))
30		difid 4375	. . . . . . 7 ⊢ (𝑋 ∖ 𝑋) = ∅
31	29, 30	eqtrdi 2792	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ 𝐵) = ∅)
32	11, 14, 31	3eqtr3d 2784	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐴 = ∅)
33	32	ex 412	. . . 4 ⊢ (𝜑 → ((𝑋 ∖ 𝐴) = 𝐵 → 𝐴 = ∅))
34	9, 33	sylbid 240	. . 3 ⊢ (𝜑 → ((𝐴 ∪ 𝐵) = 𝑋 → 𝐴 = ∅))
35	34	necon3d 2960	. 2 ⊢ (𝜑 → (𝐴 ≠ ∅ → (𝐴 ∪ 𝐵) ≠ 𝑋))
36	1, 35	mpd 15	1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ≠ 𝑋)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107   ≠ wne 2939   ∖ cdif 3947   ∪ cun 3948   ∩ cin 3949   ⊆ wss 3950  ∅c0 4332  {cpr 4627  ∪ cuni 4906  ‘cfv 6560  Topctop 22900  Clsdccld 23025  Conncconn 23420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-top 22901  df-cld 23028  df-conn 23421

This theorem is referenced by:  dfconn2  23428