Theorem conndisj 23456

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 4896	. . . . . . 7 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽)
4	2, 3	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽)
5		isconn.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
6	4, 5	sseqtrrdi 3977	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
7		conndisj.6	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
8		uneqdifeq 4445	. . . . 5 ⊢ ((𝐴 ⊆ 𝑋 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∪ 𝐵) = 𝑋 ↔ (𝑋 ∖ 𝐴) = 𝐵))
9	6, 7, 8	syl2anc 593	. . . 4 ⊢ (𝜑 → ((𝐴 ∪ 𝐵) = 𝑋 ↔ (𝑋 ∖ 𝐴) = 𝐵))
10		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ 𝐴) = 𝐵)
11	10	difeq2d 4080	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ (𝑋 ∖ 𝐴)) = (𝑋 ∖ 𝐵))
12		dfss4 4221	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴)
13	6, 12	sylib 220	. . . . . . 7 ⊢ (𝜑 → (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴)
14	13	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴)
15		connclo.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ Conn)
16	15	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐽 ∈ Conn)
17		conndisj.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ 𝐽)
18	17	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐵 ∈ 𝐽)
19		conndisj.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ≠ ∅)
20	19	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐵 ≠ ∅)
21	5	isconn 23453	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
22	21	simplbi 500	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ Conn → 𝐽 ∈ Top)
23	15, 22	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ Top)
24	5	opncld 23073	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → (𝑋 ∖ 𝐴) ∈ (Clsd‘𝐽))
25	23, 2, 24	syl2anc 593	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ∖ 𝐴) ∈ (Clsd‘𝐽))
26	25	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ 𝐴) ∈ (Clsd‘𝐽))
27	10, 26	eqeltrrd 2862	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐵 ∈ (Clsd‘𝐽))
28	5, 16, 18, 20, 27	connclo 23455	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐵 = 𝑋)
29	28	difeq2d 4080	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ 𝐵) = (𝑋 ∖ 𝑋))
30		difid 4328	. . . . . . 7 ⊢ (𝑋 ∖ 𝑋) = ∅
31	29, 30	eqtrdi 2812	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ 𝐵) = ∅)
32	11, 14, 31	3eqtr3d 2804	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐴 = ∅)
33	32	ex 416	. . . 4 ⊢ (𝜑 → ((𝑋 ∖ 𝐴) = 𝐵 → 𝐴 = ∅))
34	9, 33	sylbid 242	. . 3 ⊢ (𝜑 → ((𝐴 ∪ 𝐵) = 𝑋 → 𝐴 = ∅))
35	34	necon3d 2977	. 2 ⊢ (𝜑 → (𝐴 ≠ ∅ → (𝐴 ∪ 𝐵) ≠ 𝑋))
36	1, 35	mpd 15	1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ≠ 𝑋)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141   ≠ wne 2956   ∖ cdif 3901   ∪ cun 3902   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  {cpr 4583  ∪ cuni 4864  ‘cfv 6517  Topctop 22933  Clsdccld 23056  Conncconn 23451

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-pow 5321  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-iota 6473  df-fun 6519  df-fv 6525  df-top 22934  df-cld 23059  df-conn 23452

This theorem is referenced by:  dfconn2  23459