Theorem connclo 23240

Description: The only nonempty clopen set of a connected topology is the whole space. (Contributed 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 ≠ ∅)
connclo.4	⊢ (𝜑 → 𝐴 ∈ (Clsd‘𝐽))

Assertion

Ref	Expression
connclo	⊢ (𝜑 → 𝐴 = 𝑋)

Proof of Theorem connclo

Step	Hyp	Ref	Expression
1		connclo.3	. . 3 ⊢ (𝜑 → 𝐴 ≠ ∅)
2	1	neneqd 2937	. 2 ⊢ (𝜑 → ¬ 𝐴 = ∅)
3		connclo.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐽)
4		connclo.4	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (Clsd‘𝐽))
5	3, 4	elind 4186	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)))
6		connclo.1	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Conn)
7		isconn.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
8	7	isconn 23238	. . . . . . 7 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
9	8	simprbi 496	. . . . . 6 ⊢ (𝐽 ∈ Conn → (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})
10	6, 9	syl 17	. . . . 5 ⊢ (𝜑 → (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})
11	5, 10	eleqtrd 2827	. . . 4 ⊢ (𝜑 → 𝐴 ∈ {∅, 𝑋})
12		elpri 4642	. . . 4 ⊢ (𝐴 ∈ {∅, 𝑋} → (𝐴 = ∅ ∨ 𝐴 = 𝑋))
13	11, 12	syl 17	. . 3 ⊢ (𝜑 → (𝐴 = ∅ ∨ 𝐴 = 𝑋))
14	13	ord 861	. 2 ⊢ (𝜑 → (¬ 𝐴 = ∅ → 𝐴 = 𝑋))
15	2, 14	mpd 15	1 ⊢ (𝜑 → 𝐴 = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 844   = wceq 1533   ∈ wcel 2098   ≠ wne 2932   ∩ cin 3939  ∅c0 4314  {cpr 4622  ∪ cuni 4899  ‘cfv 6533  Topctop 22716  Clsdccld 22841  Conncconn 23236

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-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-iota 6485  df-fv 6541  df-conn 23237

This theorem is referenced by:  conndisj  23241  cnconn  23247  connsubclo  23249  t1connperf  23261  txconn  23514  connpconn  34681  cvmliftmolem2  34728  cvmlift2lem12  34760  mblfinlem1  36981