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Theorem isopn2 17097
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 3475 . . . 4  |-  ( X 
\  S )  C_  X
2 iscld.1 . . . . 5  |-  X  = 
U. J
32iscld2 17093 . . . 4  |-  ( ( J  e.  Top  /\  ( X  \  S ) 
C_  X )  -> 
( ( X  \  S )  e.  (
Clsd `  J )  <->  ( X  \  ( X 
\  S ) )  e.  J ) )
41, 3mpan2 654 . . 3  |-  ( J  e.  Top  ->  (
( X  \  S
)  e.  ( Clsd `  J )  <->  ( X  \  ( X  \  S
) )  e.  J
) )
5 dfss4 3576 . . . . 5  |-  ( S 
C_  X  <->  ( X  \  ( X  \  S
) )  =  S )
65biimpi 188 . . . 4  |-  ( S 
C_  X  ->  ( X  \  ( X  \  S ) )  =  S )
76eleq1d 2503 . . 3  |-  ( S 
C_  X  ->  (
( X  \  ( X  \  S ) )  e.  J  <->  S  e.  J ) )
84, 7sylan9bb 682 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( X  \  S )  e.  (
Clsd `  J )  <->  S  e.  J ) )
98bicomd 194 1  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( S  e.  J  <->  ( X  \  S )  e.  ( Clsd `  J
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    \ cdif 3318    C_ wss 3321   U.cuni 4016   ` cfv 5455   Topctop 16959   Clsdccld 17081
This theorem is referenced by:  opncld  17098  iscncl  17334  1stckgen  17587  txkgen  17685  qtoprest  17750  qtopcmap  17752  stoweidlem28  27754
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-iota 5419  df-fun 5457  df-fv 5463  df-top 16964  df-cld 17084
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