MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  conclo Unicode version

Theorem conclo 17401
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 2568 . 2  |-  ( ph  ->  -.  A  =  (/) )
3 conclo.2 . . . . . 6  |-  ( ph  ->  A  e.  J )
4 conclo.4 . . . . . 6  |-  ( ph  ->  A  e.  ( Clsd `  J ) )
5 elin 3475 . . . . . 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 17399 . . . . . . 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 2465 . . . 4  |-  ( ph  ->  A  e.  { (/) ,  X } )
13 elpri 3779 . . . 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 1649    e. wcel 1717    =/= wne 2552    i^i cin 3264   (/)c0 3573   {cpr 3760   U.cuni 3959   ` cfv 5396   Topctop 16883   Clsdccld 17005   Conccon 17397
This theorem is referenced by:  conndisj  17402  cnconn  17408  consubclo  17410  t1conperf  17422  txcon  17644  conpcon  24703  cvmliftmolem2  24750  cvmlift2lem12  24782
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-iota 5360  df-fv 5404  df-con 17398
  Copyright terms: Public domain W3C validator