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Theorem cmclsopn 16815
Description: The complement of a closure is open. (Contributed by NM, 11-Sep-2006.)
Hypothesis
Ref Expression
clscld.1  |-  X  = 
U. J
Assertion
Ref Expression
cmclsopn  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( X  \  (
( cls `  J
) `  S )
)  e.  J )

Proof of Theorem cmclsopn
StepHypRef Expression
1 clscld.1 . . . 4  |-  X  = 
U. J
21clsval2 16803 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  ( X  \ 
( ( int `  J
) `  ( X  \  S ) ) ) )
32difeq2d 3307 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( X  \  (
( cls `  J
) `  S )
)  =  ( X 
\  ( X  \ 
( ( int `  J
) `  ( X  \  S ) ) ) ) )
4 difss 3316 . . . . . . 7  |-  ( X 
\  S )  C_  X
51ntropn 16802 . . . . . . 7  |-  ( ( J  e.  Top  /\  ( X  \  S ) 
C_  X )  -> 
( ( int `  J
) `  ( X  \  S ) )  e.  J )
64, 5mpan2 652 . . . . . 6  |-  ( J  e.  Top  ->  (
( int `  J
) `  ( X  \  S ) )  e.  J )
71eltopss 16669 . . . . . 6  |-  ( ( J  e.  Top  /\  ( ( int `  J
) `  ( X  \  S ) )  e.  J )  ->  (
( int `  J
) `  ( X  \  S ) )  C_  X )
86, 7mpdan 649 . . . . 5  |-  ( J  e.  Top  ->  (
( int `  J
) `  ( X  \  S ) )  C_  X )
9 dfss4 3416 . . . . 5  |-  ( ( ( int `  J
) `  ( X  \  S ) )  C_  X 
<->  ( X  \  ( X  \  ( ( int `  J ) `  ( X  \  S ) ) ) )  =  ( ( int `  J
) `  ( X  \  S ) ) )
108, 9sylib 188 . . . 4  |-  ( J  e.  Top  ->  ( X  \  ( X  \ 
( ( int `  J
) `  ( X  \  S ) ) ) )  =  ( ( int `  J ) `
 ( X  \  S ) ) )
1110, 6eqeltrd 2370 . . 3  |-  ( J  e.  Top  ->  ( X  \  ( X  \ 
( ( int `  J
) `  ( X  \  S ) ) ) )  e.  J )
1211adantr 451 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( X  \  ( X  \  ( ( int `  J ) `  ( X  \  S ) ) ) )  e.  J
)
133, 12eqeltrd 2370 1  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( X  \  (
( cls `  J
) `  S )
)  e.  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    \ cdif 3162    C_ wss 3165   U.cuni 3843   ` cfv 5271   Topctop 16647   intcnt 16770   clsccl 16771
This theorem is referenced by:  elcls  16826
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-top 16652  df-cld 16772  df-ntr 16773  df-cls 16774
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