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Theorem iscon2 20207
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 20206 . 2  |-  ( J  e.  Con  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J
) )  =  { (/)
,  X } ) )
3 0opn 19705 . . . . . . 7  |-  ( J  e.  Top  ->  (/)  e.  J
)
4 0cld 19831 . . . . . . 7  |-  ( J  e.  Top  ->  (/)  e.  (
Clsd `  J )
)
53, 4elind 3627 . . . . . 6  |-  ( J  e.  Top  ->  (/)  e.  ( J  i^i  ( Clsd `  J ) ) )
61topopn 19707 . . . . . . 7  |-  ( J  e.  Top  ->  X  e.  J )
71topcld 19828 . . . . . . 7  |-  ( J  e.  Top  ->  X  e.  ( Clsd `  J
) )
86, 7elind 3627 . . . . . 6  |-  ( J  e.  Top  ->  X  e.  ( J  i^i  ( Clsd `  J ) ) )
9 prssi 4128 . . . . . 6  |-  ( (
(/)  e.  ( J  i^i  ( Clsd `  J
) )  /\  X  e.  ( J  i^i  ( Clsd `  J ) ) )  ->  { (/) ,  X }  C_  ( J  i^i  ( Clsd `  J )
) )
105, 8, 9syl2anc 659 . . . . 5  |-  ( J  e.  Top  ->  { (/) ,  X }  C_  ( J  i^i  ( Clsd `  J
) ) )
1110biantrud 505 . . . 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 )
) ) ) )
12 eqss 3457 . . . 4  |-  ( ( J  i^i  ( Clsd `  J ) )  =  { (/) ,  X }  <->  ( ( J  i^i  ( Clsd `  J ) ) 
C_  { (/) ,  X }  /\  { (/) ,  X }  C_  ( J  i^i  ( Clsd `  J )
) ) )
1311, 12syl6rbbr 264 . . 3  |-  ( J  e.  Top  ->  (
( J  i^i  ( Clsd `  J ) )  =  { (/) ,  X } 
<->  ( J  i^i  ( Clsd `  J ) ) 
C_  { (/) ,  X } ) )
1413pm5.32i 635 . 2  |-  ( ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J ) )  =  { (/) ,  X } )  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J
) )  C_  { (/) ,  X } ) )
152, 14bitri 249 1  |-  ( J  e.  Con  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J
) )  C_  { (/) ,  X } ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842    i^i cin 3413    C_ wss 3414   (/)c0 3738   {cpr 3974   U.cuni 4191   ` cfv 5569   Topctop 19686   Clsdccld 19809   Conccon 20204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-iota 5533  df-fun 5571  df-fv 5577  df-top 19691  df-cld 19812  df-con 20205
This theorem is referenced by:  indiscon  20211  dfcon2  20212  cnconn  20215  txcon  20482  filcon  20676  onsucconi  30669
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