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Theorem connclo 21266
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 2828 . 2 (𝜑 → ¬ 𝐴 = ∅)
3 connclo.2 . . . . . 6 (𝜑𝐴𝐽)
4 connclo.4 . . . . . 6 (𝜑𝐴 ∈ (Clsd‘𝐽))
53, 4elind 3831 . . . . 5 (𝜑𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)))
6 connclo.1 . . . . . 6 (𝜑𝐽 ∈ Conn)
7 isconn.1 . . . . . . . 8 𝑋 = 𝐽
87isconn 21264 . . . . . . 7 (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
98simprbi 479 . . . . . 6 (𝐽 ∈ Conn → (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})
106, 9syl 17 . . . . 5 (𝜑 → (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})
115, 10eleqtrd 2732 . . . 4 (𝜑𝐴 ∈ {∅, 𝑋})
12 elpri 4230 . . . 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 1523  wcel 2030  wne 2823  cin 3606  c0 3948  {cpr 4212   cuni 4468  cfv 5926  Topctop 20746  Clsdccld 20868  Conncconn 21262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-conn 21263
This theorem is referenced by:  conndisj  21267  cnconn  21273  connsubclo  21275  t1connperf  21287  txconn  21540  connpconn  31343  cvmliftmolem2  31390  cvmlift2lem12  31422  mblfinlem1  33576
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