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Theorem conclo 17157
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 2475 . 2  |-  ( ph  ->  -.  A  =  (/) )
3 conclo.2 . . . . . 6  |-  ( ph  ->  A  e.  J )
4 conclo.4 . . . . . 6  |-  ( ph  ->  A  e.  ( Clsd `  J ) )
5 elin 3371 . . . . . 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 17155 . . . . . . 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 2372 . . . 4  |-  ( ph  ->  A  e.  { (/) ,  X } )
13 elpri 3673 . . . 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 1632    e. wcel 1696    =/= wne 2459    i^i cin 3164   (/)c0 3468   {cpr 3654   U.cuni 3843   ` cfv 5271   Topctop 16647   Clsdccld 16769   Conccon 17153
This theorem is referenced by:  conndisj  17158  cnconn  17164  consubclo  17166  t1conperf  17178  txcon  17399  conpcon  23781  cvmliftmolem2  23828  cvmlift2lem12  23860
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-ne 2461  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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