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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 (𝜑𝐽 ∈ Con)
conclo.2 (𝜑𝐴𝐽)
conclo.3 (𝜑𝐴 ≠ ∅)
conclo.4 (𝜑𝐴 ∈ (Clsd‘𝐽))
Assertion
Ref Expression
conclo (𝜑𝐴 = 𝑋)

Proof of Theorem conclo
StepHypRef Expression
1 conclo.3 . . 3 (𝜑𝐴 ≠ ∅)
21neneqd 2787 . 2 (𝜑 → ¬ 𝐴 = ∅)
3 conclo.2 . . . . . 6 (𝜑𝐴𝐽)
4 conclo.4 . . . . . 6 (𝜑𝐴 ∈ (Clsd‘𝐽))
53, 4elind 3760 . . . . 5 (𝜑𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)))
6 conclo.1 . . . . . 6 (𝜑𝐽 ∈ Con)
7 iscon.1 . . . . . . . 8 𝑋 = 𝐽
87iscon 21026 . . . . . . 7 (𝐽 ∈ Con ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
98simprbi 479 . . . . . 6 (𝐽 ∈ Con → (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})
106, 9syl 17 . . . . 5 (𝜑 → (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})
115, 10eleqtrd 2690 . . . 4 (𝜑𝐴 ∈ {∅, 𝑋})
12 elpri 4145 . . . 4 (𝐴 ∈ {∅, 𝑋} → (𝐴 = ∅ ∨ 𝐴 = 𝑋))
1311, 12syl 17 . . 3 (𝜑 → (𝐴 = ∅ ∨ 𝐴 = 𝑋))
1413ord 391 . 2 (𝜑 → (¬ 𝐴 = ∅ → 𝐴 = 𝑋))
152, 14mpd 15 1 (𝜑𝐴 = 𝑋)
 Colors of variables: wff setvar class 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
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