Theorem conndisj 23534

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 4900	. . . . . . 7 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽)
4	2, 3	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽)
5		isconn.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
6	4, 5	sseqtrrdi 3980	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
7		conndisj.6	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
8		uneqdifeq 4449	. . . . 5 ⊢ ((𝐴 ⊆ 𝑋 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∪ 𝐵) = 𝑋 ↔ (𝑋 ∖ 𝐴) = 𝐵))
9	6, 7, 8	syl2anc 595	. . . 4 ⊢ (𝜑 → ((𝐴 ∪ 𝐵) = 𝑋 ↔ (𝑋 ∖ 𝐴) = 𝐵))
10		simpr 489	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ 𝐴) = 𝐵)
11	10	difeq2d 4083	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ (𝑋 ∖ 𝐴)) = (𝑋 ∖ 𝐵))
12		dfss4 4224	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴)
13	6, 12	sylib 221	. . . . . . 7 ⊢ (𝜑 → (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴)
14	13	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴)
15		connclo.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ Conn)
16	15	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐽 ∈ Conn)
17		conndisj.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ 𝐽)
18	17	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐵 ∈ 𝐽)
19		conndisj.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ≠ ∅)
20	19	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐵 ≠ ∅)
21	5	isconn 23531	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
22	21	simplbi 501	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ Conn → 𝐽 ∈ Top)
23	15, 22	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ Top)
24	5	opncld 23151	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → (𝑋 ∖ 𝐴) ∈ (Clsd‘𝐽))
25	23, 2, 24	syl2anc 595	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ∖ 𝐴) ∈ (Clsd‘𝐽))
26	25	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ 𝐴) ∈ (Clsd‘𝐽))
27	10, 26	eqeltrrd 2866	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐵 ∈ (Clsd‘𝐽))
28	5, 16, 18, 20, 27	connclo 23533	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐵 = 𝑋)
29	28	difeq2d 4083	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ 𝐵) = (𝑋 ∖ 𝑋))
30		difid 4332	. . . . . . 7 ⊢ (𝑋 ∖ 𝑋) = ∅
31	29, 30	eqtrdi 2816	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → (𝑋 ∖ 𝐵) = ∅)
32	11, 14, 31	3eqtr3d 2808	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∖ 𝐴) = 𝐵) → 𝐴 = ∅)
33	32	ex 417	. . . 4 ⊢ (𝜑 → ((𝑋 ∖ 𝐴) = 𝐵 → 𝐴 = ∅))
34	9, 33	sylbid 243	. . 3 ⊢ (𝜑 → ((𝐴 ∪ 𝐵) = 𝑋 → 𝐴 = ∅))
35	34	necon3d 2981	. 2 ⊢ (𝜑 → (𝐴 ≠ ∅ → (𝐴 ∪ 𝐵) ≠ 𝑋))
36	1, 35	mpd 16	1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ≠ 𝑋)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145   ≠ wne 2960   ∖ cdif 3904   ∪ cun 3905   ∩ cin 3906   ⊆ wss 3907  ∅c0 4288  {cpr 4587  ∪ cuni 4868  ‘cfv 6525  Topctop 23011  Clsdccld 23134  Conncconn 23529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-pow 5327  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-iota 6481  df-fun 6527  df-fv 6533  df-top 23012  df-cld 23137  df-conn 23530

This theorem is referenced by:  dfconn2  23537