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Theorem connclo 23423
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 2945 . 2 (𝜑 → ¬ 𝐴 = ∅)
3 connclo.2 . . . . . 6 (𝜑𝐴𝐽)
4 connclo.4 . . . . . 6 (𝜑𝐴 ∈ (Clsd‘𝐽))
53, 4elind 4200 . . . . 5 (𝜑𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)))
6 connclo.1 . . . . . 6 (𝜑𝐽 ∈ Conn)
7 isconn.1 . . . . . . . 8 𝑋 = 𝐽
87isconn 23421 . . . . . . 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 2843 . . . 4 (𝜑𝐴 ∈ {∅, 𝑋})
12 elpri 4649 . . . 4 (𝐴 ∈ {∅, 𝑋} → (𝐴 = ∅ ∨ 𝐴 = 𝑋))
1311, 12syl 17 . . 3 (𝜑 → (𝐴 = ∅ ∨ 𝐴 = 𝑋))
1413ord 865 . 2 (𝜑 → (¬ 𝐴 = ∅ → 𝐴 = 𝑋))
152, 14mpd 15 1 (𝜑𝐴 = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 848   = wceq 1540  wcel 2108  wne 2940  cin 3950  c0 4333  {cpr 4628   cuni 4907  cfv 6561  Topctop 22899  Clsdccld 23024  Conncconn 23419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-conn 23420
This theorem is referenced by:  conndisj  23424  cnconn  23430  connsubclo  23432  t1connperf  23444  txconn  23697  connpconn  35240  cvmliftmolem2  35287  cvmlift2lem12  35319  mblfinlem1  37664
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