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Theorem conntop 23426
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 2736 . . 3 𝐽 = 𝐽
21isconn 23422 . 2 (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝐽}))
32simplbi 497 1 (𝐽 ∈ Conn → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  cin 3949  c0 4332  {cpr 4627   cuni 4906  cfv 6560  Topctop 22900  Clsdccld 23025  Conncconn 23420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-conn 23421
This theorem is referenced by:  conncompss  23442  txconn  23698  qtopconn  23718  ufildr  23940  connpconn  35241  cvmliftmolem1  35287  cvmliftmolem2  35288  ordtopconn  36441
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