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Theorem iscon2 19678
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 19677 . 2  |-  ( J  e.  Con  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J
) )  =  { (/)
,  X } ) )
3 0opn 19177 . . . . . . 7  |-  ( J  e.  Top  ->  (/)  e.  J
)
4 0cld 19302 . . . . . . 7  |-  ( J  e.  Top  ->  (/)  e.  (
Clsd `  J )
)
53, 4elind 3688 . . . . . 6  |-  ( J  e.  Top  ->  (/)  e.  ( J  i^i  ( Clsd `  J ) ) )
61topopn 19179 . . . . . . 7  |-  ( J  e.  Top  ->  X  e.  J )
71topcld 19299 . . . . . . 7  |-  ( J  e.  Top  ->  X  e.  ( Clsd `  J
) )
86, 7elind 3688 . . . . . 6  |-  ( J  e.  Top  ->  X  e.  ( J  i^i  ( Clsd `  J ) ) )
9 prssi 4183 . . . . . 6  |-  ( (
(/)  e.  ( J  i^i  ( Clsd `  J
) )  /\  X  e.  ( J  i^i  ( Clsd `  J ) ) )  ->  { (/) ,  X }  C_  ( J  i^i  ( Clsd `  J )
) )
105, 8, 9syl2anc 661 . . . . 5  |-  ( J  e.  Top  ->  { (/) ,  X }  C_  ( J  i^i  ( Clsd `  J
) ) )
1110biantrud 507 . . . 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 3519 . . . 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 637 . 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 369    = wceq 1379    e. wcel 1767    i^i cin 3475    C_ wss 3476   (/)c0 3785   {cpr 4029   U.cuni 4245   ` cfv 5586   Topctop 19158   Clsdccld 19280   Conccon 19675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-top 19163  df-cld 19283  df-con 19676
This theorem is referenced by:  indiscon  19682  dfcon2  19683  cnconn  19686  txcon  19922  filcon  20116  onsucconi  29476
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