Theorem conndisj 23340

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 4944	. . . . . . 7 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽)
4	2, 3	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽)
5		isconn.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
6	4, 5	sseqtrrdi 4033	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
7		conndisj.6	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
8		uneqdifeq 4496	. . . . 5 ⊢ ((𝐴 ⊆ 𝑋 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∪ 𝐵) = 𝑋 ↔ (𝑋 ∖ 𝐴) = 𝐵))
9	6, 7, 8	syl2anc 582	. . . 4 ⊢ (𝜑 → ((𝐴 ∪ 𝐵) = 𝑋 ↔ (𝑋 ∖ 𝐴) = 𝐵))
10		simpr 483	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ 𝐴) = 𝐵)
11	10	difeq2d 4122	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ (𝑋 ∖ 𝐴)) = (𝑋 ∖ 𝐵))
12		dfss4 4261	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴)
13	6, 12	sylib 217	. . . . . . 7 ⊢ (𝜑 → (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴)
14	13	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴)
15		connclo.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ Conn)
16	15	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐽 ∈ Conn)
17		conndisj.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ 𝐽)
18	17	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐵 ∈ 𝐽)
19		conndisj.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ≠ ∅)
20	19	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐵 ≠ ∅)
21	5	isconn 23337	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
22	21	simplbi 496	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ Conn → 𝐽 ∈ Top)
23	15, 22	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ Top)
24	5	opncld 22957	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → (𝑋 ∖ 𝐴) ∈ (Clsd‘𝐽))
25	23, 2, 24	syl2anc 582	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ∖ 𝐴) ∈ (Clsd‘𝐽))
26	25	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ 𝐴) ∈ (Clsd‘𝐽))
27	10, 26	eqeltrrd 2830	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐵 ∈ (Clsd‘𝐽))
28	5, 16, 18, 20, 27	connclo 23339	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐵 = 𝑋)
29	28	difeq2d 4122	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ 𝐵) = (𝑋 ∖ 𝑋))
30		difid 4374	. . . . . . 7 ⊢ (𝑋 ∖ 𝑋) = ∅
31	29, 30	eqtrdi 2784	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ 𝐵) = ∅)
32	11, 14, 31	3eqtr3d 2776	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐴 = ∅)
33	32	ex 411	. . . 4 ⊢ (𝜑 → ((𝑋 ∖ 𝐴) = 𝐵 → 𝐴 = ∅))
34	9, 33	sylbid 239	. . 3 ⊢ (𝜑 → ((𝐴 ∪ 𝐵) = 𝑋 → 𝐴 = ∅))
35	34	necon3d 2958	. 2 ⊢ (𝜑 → (𝐴 ≠ ∅ → (𝐴 ∪ 𝐵) ≠ 𝑋))
36	1, 35	mpd 15	1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ≠ 𝑋)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ≠ wne 2937   ∖ cdif 3946   ∪ cun 3947   ∩ cin 3948   ⊆ wss 3949  ∅c0 4326  {cpr 4634  ∪ cuni 4912  ‘cfv 6553  Topctop 22815  Clsdccld 22940  Conncconn 23335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-top 22816  df-cld 22943  df-conn 23336

This theorem is referenced by:  dfconn2  23343