MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  connclo Structured version   Visualization version   GIF version

Theorem connclo 23139
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 4194 . . . . 5 (𝜑𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)))
6 connclo.1 . . . . . 6 (𝜑𝐽 ∈ Conn)
7 isconn.1 . . . . . . . 8 𝑋 = 𝐽
87isconn 23137 . . . . . . 7 (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
98simprbi 497 . . . . . 6 (𝐽 ∈ Conn → (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})
106, 9syl 17 . . . . 5 (𝜑 → (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})
115, 10eleqtrd 2835 . . . 4 (𝜑𝐴 ∈ {∅, 𝑋})
12 elpri 4650 . . . 4 (𝐴 ∈ {∅, 𝑋} → (𝐴 = ∅ ∨ 𝐴 = 𝑋))
1311, 12syl 17 . . 3 (𝜑 → (𝐴 = ∅ ∨ 𝐴 = 𝑋))
1413ord 862 . 2 (𝜑 → (¬ 𝐴 = ∅ → 𝐴 = 𝑋))
152, 14mpd 15 1 (𝜑𝐴 = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 845   = wceq 1541  wcel 2106  wne 2940  cin 3947  c0 4322  {cpr 4630   cuni 4908  cfv 6543  Topctop 22615  Clsdccld 22740  Conncconn 23135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-conn 23136
This theorem is referenced by:  conndisj  23140  cnconn  23146  connsubclo  23148  t1connperf  23160  txconn  23413  connpconn  34512  cvmliftmolem2  34559  cvmlift2lem12  34591  mblfinlem1  36828
  Copyright terms: Public domain W3C validator