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Theorem connclo 21420
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 2937 . 2 (𝜑 → ¬ 𝐴 = ∅)
3 connclo.2 . . . . . 6 (𝜑𝐴𝐽)
4 connclo.4 . . . . . 6 (𝜑𝐴 ∈ (Clsd‘𝐽))
53, 4elind 3941 . . . . 5 (𝜑𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)))
6 connclo.1 . . . . . 6 (𝜑𝐽 ∈ Conn)
7 isconn.1 . . . . . . . 8 𝑋 = 𝐽
87isconn 21418 . . . . . . 7 (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
98simprbi 483 . . . . . 6 (𝐽 ∈ Conn → (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})
106, 9syl 17 . . . . 5 (𝜑 → (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})
115, 10eleqtrd 2841 . . . 4 (𝜑𝐴 ∈ {∅, 𝑋})
12 elpri 4342 . . . 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 1632  wcel 2139  wne 2932  cin 3714  c0 4058  {cpr 4323   cuni 4588  cfv 6049  Topctop 20900  Clsdccld 21022  Conncconn 21416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-conn 21417
This theorem is referenced by:  conndisj  21421  cnconn  21427  connsubclo  21429  t1connperf  21441  txconn  21694  connpconn  31524  cvmliftmolem2  31571  cvmlift2lem12  31603  mblfinlem1  33759
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