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Theorem conclo 20082
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 2656 . 2  |-  ( ph  ->  -.  A  =  (/) )
3 conclo.2 . . . . . 6  |-  ( ph  ->  A  e.  J )
4 conclo.4 . . . . . 6  |-  ( ph  ->  A  e.  ( Clsd `  J ) )
53, 4elind 3674 . . . . 5  |-  ( ph  ->  A  e.  ( J  i^i  ( Clsd `  J
) ) )
6 conclo.1 . . . . . 6  |-  ( ph  ->  J  e.  Con )
7 iscon.1 . . . . . . . 8  |-  X  = 
U. J
87iscon 20080 . . . . . . 7  |-  ( J  e.  Con  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J
) )  =  { (/)
,  X } ) )
98simprbi 462 . . . . . 6  |-  ( J  e.  Con  ->  ( J  i^i  ( Clsd `  J
) )  =  { (/)
,  X } )
106, 9syl 16 . . . . 5  |-  ( ph  ->  ( J  i^i  ( Clsd `  J ) )  =  { (/) ,  X } )
115, 10eleqtrd 2544 . . . 4  |-  ( ph  ->  A  e.  { (/) ,  X } )
12 elpri 4036 . . . 4  |-  ( A  e.  { (/) ,  X }  ->  ( A  =  (/)  \/  A  =  X ) )
1311, 12syl 16 . . 3  |-  ( ph  ->  ( A  =  (/)  \/  A  =  X ) )
1413ord 375 . 2  |-  ( ph  ->  ( -.  A  =  (/)  ->  A  =  X ) )
152, 14mpd 15 1  |-  ( ph  ->  A  =  X )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 366    = wceq 1398    e. wcel 1823    =/= wne 2649    i^i cin 3460   (/)c0 3783   {cpr 4018   U.cuni 4235   ` cfv 5570   Topctop 19561   Clsdccld 19684   Conccon 20078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-con 20079
This theorem is referenced by:  conndisj  20083  cnconn  20089  consubclo  20091  t1conperf  20103  txcon  20356  conpcon  28944  cvmliftmolem2  28991  cvmlift2lem12  29023  mblfinlem1  30291
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