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

Theorem conntop 22921
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 2733 . . 3 𝐽 = 𝐽
21isconn 22917 . 2 (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝐽}))
32simplbi 499 1 (𝐽 ∈ Conn → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cin 3948  c0 4323  {cpr 4631   cuni 4909  cfv 6544  Topctop 22395  Clsdccld 22520  Conncconn 22915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-conn 22916
This theorem is referenced by:  conncompss  22937  txconn  23193  qtopconn  23213  ufildr  23435  connpconn  34226  cvmliftmolem1  34272  cvmliftmolem2  34273  ordtopconn  35324
  Copyright terms: Public domain W3C validator