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Theorem isopn2 19294
Description: A subset of the underlying set of a topology is open iff its complement is closed. (Contributed by NM, 4-Oct-2006.)
Hypothesis
Ref Expression
iscld.1  |-  X  = 
U. J
Assertion
Ref Expression
isopn2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( S  e.  J  <->  ( X  \  S )  e.  ( Clsd `  J
) ) )

Proof of Theorem isopn2
StepHypRef Expression
1 difss 3626 . . . 4  |-  ( X 
\  S )  C_  X
2 iscld.1 . . . . 5  |-  X  = 
U. J
32iscld2 19290 . . . 4  |-  ( ( J  e.  Top  /\  ( X  \  S ) 
C_  X )  -> 
( ( X  \  S )  e.  (
Clsd `  J )  <->  ( X  \  ( X 
\  S ) )  e.  J ) )
41, 3mpan2 671 . . 3  |-  ( J  e.  Top  ->  (
( X  \  S
)  e.  ( Clsd `  J )  <->  ( X  \  ( X  \  S
) )  e.  J
) )
5 dfss4 3727 . . . . 5  |-  ( S 
C_  X  <->  ( X  \  ( X  \  S
) )  =  S )
65biimpi 194 . . . 4  |-  ( S 
C_  X  ->  ( X  \  ( X  \  S ) )  =  S )
76eleq1d 2531 . . 3  |-  ( S 
C_  X  ->  (
( X  \  ( X  \  S ) )  e.  J  <->  S  e.  J ) )
84, 7sylan9bb 699 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( X  \  S )  e.  (
Clsd `  J )  <->  S  e.  J ) )
98bicomd 201 1  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( S  e.  J  <->  ( X  \  S )  e.  ( Clsd `  J
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762    \ cdif 3468    C_ wss 3471   U.cuni 4240   ` cfv 5581   Topctop 19156   Clsdccld 19278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-iota 5544  df-fun 5583  df-fv 5589  df-top 19161  df-cld 19281
This theorem is referenced by:  opncld  19295  iscncl  19531  1stckgen  19785  txkgen  19883  qtoprest  19948  qtopcmap  19950  stoweidlem28  31285
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