Theorem conclo 21028

Description: The only nonempty clopen set of a connected topology is the whole space. (Contributed by Mario Carneiro, 10-Mar-2015.)

Hypotheses

Ref	Expression
iscon.1	⊢ 𝑋 = ∪ 𝐽
conclo.1	⊢ (𝜑 → 𝐽 ∈ Con)
conclo.2	⊢ (𝜑 → 𝐴 ∈ 𝐽)
conclo.3	⊢ (𝜑 → 𝐴 ≠ ∅)
conclo.4	⊢ (𝜑 → 𝐴 ∈ (Clsd‘𝐽))

Assertion

Ref	Expression
conclo	⊢ (𝜑 → 𝐴 = 𝑋)

Proof of Theorem conclo

Step	Hyp	Ref	Expression
1		conclo.3	. . 3 ⊢ (𝜑 → 𝐴 ≠ ∅)
2	1	neneqd 2787	. 2 ⊢ (𝜑 → ¬ 𝐴 = ∅)
3		conclo.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐽)
4		conclo.4	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (Clsd‘𝐽))
5	3, 4	elind 3760	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)))
6		conclo.1	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Con)
7		iscon.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
8	7	iscon 21026	. . . . . . 7 ⊢ (𝐽 ∈ Con ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
9	8	simprbi 479	. . . . . 6 ⊢ (𝐽 ∈ Con → (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})
10	6, 9	syl 17	. . . . 5 ⊢ (𝜑 → (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})
11	5, 10	eleqtrd 2690	. . . 4 ⊢ (𝜑 → 𝐴 ∈ {∅, 𝑋})
12		elpri 4145	. . . 4 ⊢ (𝐴 ∈ {∅, 𝑋} → (𝐴 = ∅ ∨ 𝐴 = 𝑋))
13	11, 12	syl 17	. . 3 ⊢ (𝜑 → (𝐴 = ∅ ∨ 𝐴 = 𝑋))
14	13	ord 391	. 2 ⊢ (𝜑 → (¬ 𝐴 = ∅ → 𝐴 = 𝑋))
15	2, 14	mpd 15	1 ⊢ (𝜑 → 𝐴 = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∩ cin 3539  ∅c0 3874  {cpr 4127  ∪ cuni 4372  ‘cfv 5804  Topctop 20517  Clsdccld 20630  Conccon 21024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-con 21025

This theorem is referenced by:  conndisj  21029  cnconn  21035  consubclo  21037  t1conperf  21049  txcon  21302  conpcon  30471  cvmliftmolem2  30518  cvmlift2lem12  30550  mblfinlem1  32616