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Theorem conclo 17470
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 2614 . 2  |-  ( ph  ->  -.  A  =  (/) )
3 conclo.2 . . . . . 6  |-  ( ph  ->  A  e.  J )
4 conclo.4 . . . . . 6  |-  ( ph  ->  A  e.  ( Clsd `  J ) )
5 elin 3522 . . . . . 6  |-  ( A  e.  ( J  i^i  ( Clsd `  J )
)  <->  ( A  e.  J  /\  A  e.  ( Clsd `  J
) ) )
63, 4, 5sylanbrc 646 . . . . 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 17468 . . . . . . 7  |-  ( J  e.  Con  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J
) )  =  { (/)
,  X } ) )
109simprbi 451 . . . . . 6  |-  ( J  e.  Con  ->  ( J  i^i  ( Clsd `  J
) )  =  { (/)
,  X } )
117, 10syl 16 . . . . 5  |-  ( ph  ->  ( J  i^i  ( Clsd `  J ) )  =  { (/) ,  X } )
126, 11eleqtrd 2511 . . . 4  |-  ( ph  ->  A  e.  { (/) ,  X } )
13 elpri 3826 . . . 4  |-  ( A  e.  { (/) ,  X }  ->  ( A  =  (/)  \/  A  =  X ) )
1412, 13syl 16 . . 3  |-  ( ph  ->  ( A  =  (/)  \/  A  =  X ) )
1514ord 367 . 2  |-  ( ph  ->  ( -.  A  =  (/)  ->  A  =  X ) )
162, 15mpd 15 1  |-  ( ph  ->  A  =  X )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    = wceq 1652    e. wcel 1725    =/= wne 2598    i^i cin 3311   (/)c0 3620   {cpr 3807   U.cuni 4007   ` cfv 5446   Topctop 16950   Clsdccld 17072   Conccon 17466
This theorem is referenced by:  conndisj  17471  cnconn  17477  consubclo  17479  t1conperf  17491  txcon  17713  conpcon  24914  cvmliftmolem2  24961  cvmlift2lem12  24993  mblfinlem  26234
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-con 17467
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