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Theorem isopn2 19618
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 3545 . . . 4  |-  ( X 
\  S )  C_  X
2 iscld.1 . . . . 5  |-  X  = 
U. J
32iscld2 19614 . . . 4  |-  ( ( J  e.  Top  /\  ( X  \  S ) 
C_  X )  -> 
( ( X  \  S )  e.  (
Clsd `  J )  <->  ( X  \  ( X 
\  S ) )  e.  J ) )
41, 3mpan2 669 . . 3  |-  ( J  e.  Top  ->  (
( X  \  S
)  e.  ( Clsd `  J )  <->  ( X  \  ( X  \  S
) )  e.  J
) )
5 dfss4 3657 . . . . 5  |-  ( S 
C_  X  <->  ( X  \  ( X  \  S
) )  =  S )
65biimpi 194 . . . 4  |-  ( S 
C_  X  ->  ( X  \  ( X  \  S ) )  =  S )
76eleq1d 2451 . . 3  |-  ( S 
C_  X  ->  (
( X  \  ( X  \  S ) )  e.  J  <->  S  e.  J ) )
84, 7sylan9bb 697 . 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 367    = wceq 1399    e. wcel 1826    \ cdif 3386    C_ wss 3389   U.cuni 4163   ` cfv 5496   Topctop 19479   Clsdccld 19602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5460  df-fun 5498  df-fv 5504  df-top 19484  df-cld 19605
This theorem is referenced by:  opncld  19619  iscncl  19856  1stckgen  20140  txkgen  20238  qtoprest  20303  qtopcmap  20305  stoweidlem28  31976
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