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Theorem iscon2 17477
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 17476 . 2  |-  ( J  e.  Con  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J
) )  =  { (/)
,  X } ) )
3 0opn 16977 . . . . . . 7  |-  ( J  e.  Top  ->  (/)  e.  J
)
4 0cld 17102 . . . . . . 7  |-  ( J  e.  Top  ->  (/)  e.  (
Clsd `  J )
)
5 elin 3530 . . . . . . 7  |-  ( (/)  e.  ( J  i^i  ( Clsd `  J ) )  <-> 
( (/)  e.  J  /\  (/) 
e.  ( Clsd `  J
) ) )
63, 4, 5sylanbrc 646 . . . . . 6  |-  ( J  e.  Top  ->  (/)  e.  ( J  i^i  ( Clsd `  J ) ) )
71topopn 16979 . . . . . . 7  |-  ( J  e.  Top  ->  X  e.  J )
81topcld 17099 . . . . . . 7  |-  ( J  e.  Top  ->  X  e.  ( Clsd `  J
) )
9 elin 3530 . . . . . . 7  |-  ( X  e.  ( J  i^i  ( Clsd `  J )
)  <->  ( X  e.  J  /\  X  e.  ( Clsd `  J
) ) )
107, 8, 9sylanbrc 646 . . . . . 6  |-  ( J  e.  Top  ->  X  e.  ( J  i^i  ( Clsd `  J ) ) )
11 prssi 3954 . . . . . 6  |-  ( (
(/)  e.  ( J  i^i  ( Clsd `  J
) )  /\  X  e.  ( J  i^i  ( Clsd `  J ) ) )  ->  { (/) ,  X }  C_  ( J  i^i  ( Clsd `  J )
) )
126, 10, 11syl2anc 643 . . . . 5  |-  ( J  e.  Top  ->  { (/) ,  X }  C_  ( J  i^i  ( Clsd `  J
) ) )
1312biantrud 494 . . . 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 3363 . . . 4  |-  ( ( J  i^i  ( Clsd `  J ) )  =  { (/) ,  X }  <->  ( ( J  i^i  ( Clsd `  J ) ) 
C_  { (/) ,  X }  /\  { (/) ,  X }  C_  ( J  i^i  ( Clsd `  J )
) ) )
1513, 14syl6rbbr 256 . . 3  |-  ( J  e.  Top  ->  (
( J  i^i  ( Clsd `  J ) )  =  { (/) ,  X } 
<->  ( J  i^i  ( Clsd `  J ) ) 
C_  { (/) ,  X } ) )
1615pm5.32i 619 . 2  |-  ( ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J ) )  =  { (/) ,  X } )  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J
) )  C_  { (/) ,  X } ) )
172, 16bitri 241 1  |-  ( J  e.  Con  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J
) )  C_  { (/) ,  X } ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    i^i cin 3319    C_ wss 3320   (/)c0 3628   {cpr 3815   U.cuni 4015   ` cfv 5454   Topctop 16958   Clsdccld 17080   Conccon 17474
This theorem is referenced by:  indiscon  17481  dfcon2  17482  cnconn  17485  txcon  17721  filcon  17915  onsucconi  26187
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
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-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-top 16963  df-cld 17083  df-con 17475
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