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

Theorem conntop 22025
 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 2824 . . 3 𝐽 = 𝐽
21isconn 22021 . 2 (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝐽}))
32simplbi 501 1 (𝐽 ∈ Conn → 𝐽 ∈ Top)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2115   ∩ cin 3918  ∅c0 4276  {cpr 4552  ∪ cuni 4824  ‘cfv 6343  Topctop 21501  Clsdccld 21624  Conncconn 22019 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-iota 6302  df-fv 6351  df-conn 22020 This theorem is referenced by:  conncompss  22041  txconn  22297  qtopconn  22317  ufildr  22539  connpconn  32539  cvmliftmolem1  32585  cvmliftmolem2  32586  ordtopconn  33844
 Copyright terms: Public domain W3C validator