Theorem conndisj 23739

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 4898	. . . . . . 7 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽)
4	2, 3	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽)
5		isconn.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
6	4, 5	sseqtrrdi 3949	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
7		conndisj.6	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
8		uneqdifeq 4447	. . . . 5 ⊢ ((𝐴 ⊆ 𝑋 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∪ 𝐵) = 𝑋 ↔ (𝑋 ∖ 𝐴) = 𝐵))
9	6, 7, 8	syl2anc 596	. . . 4 ⊢ (𝜑 → ((𝐴 ∪ 𝐵) = 𝑋 ↔ (𝑋 ∖ 𝐴) = 𝐵))
10		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ 𝐴) = 𝐵)
11	10	difeq2d 4063	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ (𝑋 ∖ 𝐴)) = (𝑋 ∖ 𝐵))
12		dfss4 4214	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴)
13	6, 12	sylib 221	. . . . . . 7 ⊢ (𝜑 → (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴)
14	13	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴)
15		connclo.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ Conn)
16	15	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐽 ∈ Conn)
17		conndisj.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ 𝐽)
18	17	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐵 ∈ 𝐽)
19		conndisj.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ≠ ∅)
20	19	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐵 ≠ ∅)
21	5	isconn 23736	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
22	21	simplbi 502	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ Conn → 𝐽 ∈ Top)
23	15, 22	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ Top)
24	5	opncld 23356	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → (𝑋 ∖ 𝐴) ∈ (Clsd‘𝐽))
25	23, 2, 24	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ∖ 𝐴) ∈ (Clsd‘𝐽))
26	25	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ 𝐴) ∈ (Clsd‘𝐽))
27	10, 26	eqeltrrd 2861	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐵 ∈ (Clsd‘𝐽))
28	5, 16, 18, 20, 27	connclo 23738	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐵 = 𝑋)
29	28	difeq2d 4063	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ 𝐵) = (𝑋 ∖ 𝑋))
30		difid 4324	. . . . . . 7 ⊢ (𝑋 ∖ 𝑋) = ∅
31	29, 30	eqtrdi 2811	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ 𝐵) = ∅)
32	11, 14, 31	3eqtr3d 2803	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐴 = ∅)
33	32	ex 418	. . . 4 ⊢ (𝜑 → ((𝑋 ∖ 𝐴) = 𝐵 → 𝐴 = ∅))
34	9, 33	sylbid 243	. . 3 ⊢ (𝜑 → ((𝐴 ∪ 𝐵) = 𝑋 → 𝐴 = ∅))
35	34	necon3d 2976	. 2 ⊢ (𝜑 → (𝐴 ≠ ∅ → (𝐴 ∪ 𝐵) ≠ 𝑋))
36	1, 35	mpd 16	1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ≠ 𝑋)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955   ⊆ wss 3895   ∖ cdif 4045   ∩ cin 4089   ∪ cun 4149  ∅c0 4278  {cpr 4586  ∪ cuni 4866  ‘cfv 6538  Topctop 23216  Clsdccld 23339  Conncconn 23734

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pow 5326  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6540  df-fv 6546  df-top 23217  df-cld 23342  df-conn 23735

This theorem is used by:  dfconn2  23742