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Theorem contop 17159
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 2296 . . 3  |-  U. J  =  U. J
21iscon 17155 . 2  |-  ( J  e.  Con  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J
) )  =  { (/)
,  U. J } ) )
32simplbi 446 1  |-  ( J  e.  Con  ->  J  e.  Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    i^i cin 3164   (/)c0 3468   {cpr 3654   U.cuni 3843   ` cfv 5271   Topctop 16647   Clsdccld 16769   Conccon 17153
This theorem is referenced by:  concompss  17175  txcon  17399  qtopcon  17416  ufildr  17642  conpcon  23781  cvmliftmolem1  23827  cvmliftmolem2  23828  ordtopcon  24950
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-con 17154
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