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Theorem isopn2 16825
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 3337 . . . 4  |-  ( X 
\  S )  C_  X
2 iscld.1 . . . . 5  |-  X  = 
U. J
32iscld2 16821 . . . 4  |-  ( ( J  e.  Top  /\  ( X  \  S ) 
C_  X )  -> 
( ( X  \  S )  e.  (
Clsd `  J )  <->  ( X  \  ( X 
\  S ) )  e.  J ) )
41, 3mpan2 652 . . 3  |-  ( J  e.  Top  ->  (
( X  \  S
)  e.  ( Clsd `  J )  <->  ( X  \  ( X  \  S
) )  e.  J
) )
5 dfss4 3437 . . . . 5  |-  ( S 
C_  X  <->  ( X  \  ( X  \  S
) )  =  S )
65biimpi 186 . . . 4  |-  ( S 
C_  X  ->  ( X  \  ( X  \  S ) )  =  S )
76eleq1d 2382 . . 3  |-  ( S 
C_  X  ->  (
( X  \  ( X  \  S ) )  e.  J  <->  S  e.  J ) )
84, 7sylan9bb 680 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( X  \  S )  e.  (
Clsd `  J )  <->  S  e.  J ) )
98bicomd 192 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 176    /\ wa 358    = wceq 1633    e. wcel 1701    \ cdif 3183    C_ wss 3186   U.cuni 3864   ` cfv 5292   Topctop 16687   Clsdccld 16809
This theorem is referenced by:  opncld  16826  iscncl  17054  1stckgen  17305  txkgen  17402  qtoprest  17464  qtopcmap  17466  stoweidlem28  26925
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-iota 5256  df-fun 5294  df-fv 5300  df-top 16692  df-cld 16812
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