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Theorem contop 17481
Description: A connected topology is a topology. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 14-Dec-2013.)
Assertion
Ref Expression
contop  |-  ( J  e.  Con  ->  J  e.  Top )

Proof of Theorem contop
StepHypRef Expression
1 eqid 2437 . . 3  |-  U. J  =  U. J
21iscon 17477 . 2  |-  ( J  e.  Con  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J
) )  =  { (/)
,  U. J } ) )
32simplbi 448 1  |-  ( J  e.  Con  ->  J  e.  Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726    i^i cin 3320   (/)c0 3629   {cpr 3816   U.cuni 4016   ` cfv 5455   Topctop 16959   Clsdccld 17081   Conccon 17475
This theorem is referenced by:  concompss  17497  txcon  17722  qtopcon  17742  ufildr  17964  conpcon  24923  cvmliftmolem1  24969  cvmliftmolem2  24970  ordtopcon  26190
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-iota 5419  df-fv 5463  df-con 17476
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