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Theorem iscon2 20422
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 20421 . 2  |-  ( J  e.  Con  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J
) )  =  { (/)
,  X } ) )
3 0opn 19927 . . . . . . 7  |-  ( J  e.  Top  ->  (/)  e.  J
)
4 0cld 20046 . . . . . . 7  |-  ( J  e.  Top  ->  (/)  e.  (
Clsd `  J )
)
53, 4elind 3617 . . . . . 6  |-  ( J  e.  Top  ->  (/)  e.  ( J  i^i  ( Clsd `  J ) ) )
61topopn 19929 . . . . . . 7  |-  ( J  e.  Top  ->  X  e.  J )
71topcld 20043 . . . . . . 7  |-  ( J  e.  Top  ->  X  e.  ( Clsd `  J
) )
86, 7elind 3617 . . . . . 6  |-  ( J  e.  Top  ->  X  e.  ( J  i^i  ( Clsd `  J ) ) )
9 prssi 4127 . . . . . 6  |-  ( (
(/)  e.  ( J  i^i  ( Clsd `  J
) )  /\  X  e.  ( J  i^i  ( Clsd `  J ) ) )  ->  { (/) ,  X }  C_  ( J  i^i  ( Clsd `  J )
) )
105, 8, 9syl2anc 666 . . . . 5  |-  ( J  e.  Top  ->  { (/) ,  X }  C_  ( J  i^i  ( Clsd `  J
) ) )
1110biantrud 510 . . . 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 3446 . . . 4  |-  ( ( J  i^i  ( Clsd `  J ) )  =  { (/) ,  X }  <->  ( ( J  i^i  ( Clsd `  J ) ) 
C_  { (/) ,  X }  /\  { (/) ,  X }  C_  ( J  i^i  ( Clsd `  J )
) ) )
1311, 12syl6rbbr 268 . . 3  |-  ( J  e.  Top  ->  (
( J  i^i  ( Clsd `  J ) )  =  { (/) ,  X } 
<->  ( J  i^i  ( Clsd `  J ) ) 
C_  { (/) ,  X } ) )
1413pm5.32i 642 . 2  |-  ( ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J ) )  =  { (/) ,  X } )  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J
) )  C_  { (/) ,  X } ) )
152, 14bitri 253 1  |-  ( J  e.  Con  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J
) )  C_  { (/) ,  X } ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371    = wceq 1443    e. wcel 1886    i^i cin 3402    C_ wss 3403   (/)c0 3730   {cpr 3969   U.cuni 4197   ` cfv 5581   Topctop 19910   Clsdccld 20024   Conccon 20419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-iota 5545  df-fun 5583  df-fv 5589  df-top 19914  df-cld 20027  df-con 20420
This theorem is referenced by:  indiscon  20426  dfcon2  20427  cnconn  20430  txcon  20697  filcon  20891  onsucconi  31090
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