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Theorem conntop 23141
Description: A connected topology is a topology. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 14-Dec-2013.)
Assertion
Ref Expression
conntop (𝐽 ∈ Conn → 𝐽 ∈ Top)

Proof of Theorem conntop
StepHypRef Expression
1 eqid 2730 . . 3 𝐽 = 𝐽
21isconn 23137 . 2 (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝐽}))
32simplbi 496 1 (𝐽 ∈ Conn → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2104  cin 3946  c0 4321  {cpr 4629   cuni 4907  cfv 6542  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 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6494  df-fv 6550  df-conn 23136
This theorem is referenced by:  conncompss  23157  txconn  23413  qtopconn  23433  ufildr  23655  connpconn  34524  cvmliftmolem1  34570  cvmliftmolem2  34571  ordtopconn  35627
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