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Theorem conclo 19144
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 2651 . 2  |-  ( ph  ->  -.  A  =  (/) )
3 conclo.2 . . . . . 6  |-  ( ph  ->  A  e.  J )
4 conclo.4 . . . . . 6  |-  ( ph  ->  A  e.  ( Clsd `  J ) )
53, 4elind 3641 . . . . 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 19142 . . . . . . 7  |-  ( J  e.  Con  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J
) )  =  { (/)
,  X } ) )
98simprbi 464 . . . . . 6  |-  ( J  e.  Con  ->  ( J  i^i  ( Clsd `  J
) )  =  { (/)
,  X } )
106, 9syl 16 . . . . 5  |-  ( ph  ->  ( J  i^i  ( Clsd `  J ) )  =  { (/) ,  X } )
115, 10eleqtrd 2541 . . . 4  |-  ( ph  ->  A  e.  { (/) ,  X } )
12 elpri 3998 . . . 4  |-  ( A  e.  { (/) ,  X }  ->  ( A  =  (/)  \/  A  =  X ) )
1311, 12syl 16 . . 3  |-  ( ph  ->  ( A  =  (/)  \/  A  =  X ) )
1413ord 377 . 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 368    = wceq 1370    e. wcel 1758    =/= wne 2644    i^i cin 3428   (/)c0 3738   {cpr 3980   U.cuni 4192   ` cfv 5519   Topctop 18623   Clsdccld 18745   Conccon 19140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-iota 5482  df-fv 5527  df-con 19141
This theorem is referenced by:  conndisj  19145  cnconn  19151  consubclo  19153  t1conperf  19165  txcon  19387  conpcon  27261  cvmliftmolem2  27308  cvmlift2lem12  27340  mblfinlem1  28569
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