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Theorem connclo 23438
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 (𝜑𝐽 ∈ Conn)
connclo.2 (𝜑𝐴𝐽)
connclo.3 (𝜑𝐴 ≠ ∅)
connclo.4 (𝜑𝐴 ∈ (Clsd‘𝐽))
Assertion
Ref Expression
connclo (𝜑𝐴 = 𝑋)

Proof of Theorem connclo
StepHypRef Expression
1 connclo.3 . . 3 (𝜑𝐴 ≠ ∅)
21neneqd 2942 . 2 (𝜑 → ¬ 𝐴 = ∅)
3 connclo.2 . . . . . 6 (𝜑𝐴𝐽)
4 connclo.4 . . . . . 6 (𝜑𝐴 ∈ (Clsd‘𝐽))
53, 4elind 4209 . . . . 5 (𝜑𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)))
6 connclo.1 . . . . . 6 (𝜑𝐽 ∈ Conn)
7 isconn.1 . . . . . . . 8 𝑋 = 𝐽
87isconn 23436 . . . . . . 7 (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
98simprbi 496 . . . . . 6 (𝐽 ∈ Conn → (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})
106, 9syl 17 . . . . 5 (𝜑 → (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})
115, 10eleqtrd 2840 . . . 4 (𝜑𝐴 ∈ {∅, 𝑋})
12 elpri 4653 . . . 4 (𝐴 ∈ {∅, 𝑋} → (𝐴 = ∅ ∨ 𝐴 = 𝑋))
1311, 12syl 17 . . 3 (𝜑 → (𝐴 = ∅ ∨ 𝐴 = 𝑋))
1413ord 864 . 2 (𝜑 → (¬ 𝐴 = ∅ → 𝐴 = 𝑋))
152, 14mpd 15 1 (𝜑𝐴 = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 847   = wceq 1536  wcel 2105  wne 2937  cin 3961  c0 4338  {cpr 4632   cuni 4911  cfv 6562  Topctop 22914  Clsdccld 23039  Conncconn 23434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-iota 6515  df-fv 6570  df-conn 23435
This theorem is referenced by:  conndisj  23439  cnconn  23445  connsubclo  23447  t1connperf  23459  txconn  23712  connpconn  35219  cvmliftmolem2  35266  cvmlift2lem12  35298  mblfinlem1  37643
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