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Theorem iscon2 17156
Description: The predicate  J is a connected topology . (Contributed by Mario Carneiro, 10-Mar-2015.)
Hypothesis
Ref Expression
iscon.1  |-  X  = 
U. J
Assertion
Ref Expression
iscon2  |-  ( J  e.  Con  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J
) )  C_  { (/) ,  X } ) )

Proof of Theorem iscon2
StepHypRef Expression
1 iscon.1 . . 3  |-  X  = 
U. J
21iscon 17155 . 2  |-  ( J  e.  Con  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J
) )  =  { (/)
,  X } ) )
3 0opn 16666 . . . . . . 7  |-  ( J  e.  Top  ->  (/)  e.  J
)
4 0cld 16791 . . . . . . 7  |-  ( J  e.  Top  ->  (/)  e.  (
Clsd `  J )
)
5 elin 3371 . . . . . . 7  |-  ( (/)  e.  ( J  i^i  ( Clsd `  J ) )  <-> 
( (/)  e.  J  /\  (/) 
e.  ( Clsd `  J
) ) )
63, 4, 5sylanbrc 645 . . . . . 6  |-  ( J  e.  Top  ->  (/)  e.  ( J  i^i  ( Clsd `  J ) ) )
71topopn 16668 . . . . . . 7  |-  ( J  e.  Top  ->  X  e.  J )
81topcld 16788 . . . . . . 7  |-  ( J  e.  Top  ->  X  e.  ( Clsd `  J
) )
9 elin 3371 . . . . . . 7  |-  ( X  e.  ( J  i^i  ( Clsd `  J )
)  <->  ( X  e.  J  /\  X  e.  ( Clsd `  J
) ) )
107, 8, 9sylanbrc 645 . . . . . 6  |-  ( J  e.  Top  ->  X  e.  ( J  i^i  ( Clsd `  J ) ) )
11 prssi 3787 . . . . . 6  |-  ( (
(/)  e.  ( J  i^i  ( Clsd `  J
) )  /\  X  e.  ( J  i^i  ( Clsd `  J ) ) )  ->  { (/) ,  X }  C_  ( J  i^i  ( Clsd `  J )
) )
126, 10, 11syl2anc 642 . . . . 5  |-  ( J  e.  Top  ->  { (/) ,  X }  C_  ( J  i^i  ( Clsd `  J
) ) )
1312biantrud 493 . . . 4  |-  ( J  e.  Top  ->  (
( J  i^i  ( Clsd `  J ) ) 
C_  { (/) ,  X } 
<->  ( ( J  i^i  ( Clsd `  J )
)  C_  { (/) ,  X }  /\  { (/) ,  X }  C_  ( J  i^i  ( Clsd `  J )
) ) ) )
14 eqss 3207 . . . 4  |-  ( ( J  i^i  ( Clsd `  J ) )  =  { (/) ,  X }  <->  ( ( J  i^i  ( Clsd `  J ) ) 
C_  { (/) ,  X }  /\  { (/) ,  X }  C_  ( J  i^i  ( Clsd `  J )
) ) )
1513, 14syl6rbbr 255 . . 3  |-  ( J  e.  Top  ->  (
( J  i^i  ( Clsd `  J ) )  =  { (/) ,  X } 
<->  ( J  i^i  ( Clsd `  J ) ) 
C_  { (/) ,  X } ) )
1615pm5.32i 618 . 2  |-  ( ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J ) )  =  { (/) ,  X } )  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J
) )  C_  { (/) ,  X } ) )
172, 16bitri 240 1  |-  ( J  e.  Con  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J
) )  C_  { (/) ,  X } ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   (/)c0 3468   {cpr 3654   U.cuni 3843   ` cfv 5271   Topctop 16647   Clsdccld 16769   Conccon 17153
This theorem is referenced by:  indiscon  17160  dfcon2  17161  cnconn  17164  txcon  17399  filcon  17594  onsucconi  24948
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
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-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-top 16652  df-cld 16772  df-con 17154
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