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Theorem conclo 17141
Description: The only nonempty clopen set of a connected topology is the whole space. (Contributed by Mario Carneiro, 10-Mar-2015.)
Hypotheses
Ref Expression
iscon.1  |-  X  = 
U. J
conclo.1  |-  ( ph  ->  J  e.  Con )
conclo.2  |-  ( ph  ->  A  e.  J )
conclo.3  |-  ( ph  ->  A  =/=  (/) )
conclo.4  |-  ( ph  ->  A  e.  ( Clsd `  J ) )
Assertion
Ref Expression
conclo  |-  ( ph  ->  A  =  X )

Proof of Theorem conclo
StepHypRef Expression
1 conclo.3 . . 3  |-  ( ph  ->  A  =/=  (/) )
21neneqd 2462 . 2  |-  ( ph  ->  -.  A  =  (/) )
3 conclo.2 . . . . . 6  |-  ( ph  ->  A  e.  J )
4 conclo.4 . . . . . 6  |-  ( ph  ->  A  e.  ( Clsd `  J ) )
5 elin 3358 . . . . . 6  |-  ( A  e.  ( J  i^i  ( Clsd `  J )
)  <->  ( A  e.  J  /\  A  e.  ( Clsd `  J
) ) )
63, 4, 5sylanbrc 645 . . . . 5  |-  ( ph  ->  A  e.  ( J  i^i  ( Clsd `  J
) ) )
7 conclo.1 . . . . . 6  |-  ( ph  ->  J  e.  Con )
8 iscon.1 . . . . . . . 8  |-  X  = 
U. J
98iscon 17139 . . . . . . 7  |-  ( J  e.  Con  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J
) )  =  { (/)
,  X } ) )
109simprbi 450 . . . . . 6  |-  ( J  e.  Con  ->  ( J  i^i  ( Clsd `  J
) )  =  { (/)
,  X } )
117, 10syl 15 . . . . 5  |-  ( ph  ->  ( J  i^i  ( Clsd `  J ) )  =  { (/) ,  X } )
126, 11eleqtrd 2359 . . . 4  |-  ( ph  ->  A  e.  { (/) ,  X } )
13 elpri 3660 . . . 4  |-  ( A  e.  { (/) ,  X }  ->  ( A  =  (/)  \/  A  =  X ) )
1412, 13syl 15 . . 3  |-  ( ph  ->  ( A  =  (/)  \/  A  =  X ) )
1514ord 366 . 2  |-  ( ph  ->  ( -.  A  =  (/)  ->  A  =  X ) )
162, 15mpd 14 1  |-  ( ph  ->  A  =  X )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    = wceq 1623    e. wcel 1684    =/= wne 2446    i^i cin 3151   (/)c0 3455   {cpr 3641   U.cuni 3827   ` cfv 5255   Topctop 16631   Clsdccld 16753   Conccon 17137
This theorem is referenced by:  conndisj  17142  cnconn  17148  consubclo  17150  t1conperf  17162  txcon  17383  conpcon  23766  cvmliftmolem2  23813  cvmlift2lem12  23845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-con 17138
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