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Theorem cmclsopn 17128
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 17116 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  ( X  \ 
( ( int `  J
) `  ( X  \  S ) ) ) )
32difeq2d 3467 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( X  \  (
( cls `  J
) `  S )
)  =  ( X 
\  ( X  \ 
( ( int `  J
) `  ( X  \  S ) ) ) ) )
4 difss 3476 . . . . . . 7  |-  ( X 
\  S )  C_  X
51ntropn 17115 . . . . . . 7  |-  ( ( J  e.  Top  /\  ( X  \  S ) 
C_  X )  -> 
( ( int `  J
) `  ( X  \  S ) )  e.  J )
64, 5mpan2 654 . . . . . 6  |-  ( J  e.  Top  ->  (
( int `  J
) `  ( X  \  S ) )  e.  J )
71eltopss 16982 . . . . . 6  |-  ( ( J  e.  Top  /\  ( ( int `  J
) `  ( X  \  S ) )  e.  J )  ->  (
( int `  J
) `  ( X  \  S ) )  C_  X )
86, 7mpdan 651 . . . . 5  |-  ( J  e.  Top  ->  (
( int `  J
) `  ( X  \  S ) )  C_  X )
9 dfss4 3577 . . . . 5  |-  ( ( ( int `  J
) `  ( X  \  S ) )  C_  X 
<->  ( X  \  ( X  \  ( ( int `  J ) `  ( X  \  S ) ) ) )  =  ( ( int `  J
) `  ( X  \  S ) ) )
108, 9sylib 190 . . . 4  |-  ( J  e.  Top  ->  ( X  \  ( X  \ 
( ( int `  J
) `  ( X  \  S ) ) ) )  =  ( ( int `  J ) `
 ( X  \  S ) ) )
1110, 6eqeltrd 2512 . . 3  |-  ( J  e.  Top  ->  ( X  \  ( X  \ 
( ( int `  J
) `  ( X  \  S ) ) ) )  e.  J )
1211adantr 453 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( X  \  ( X  \  ( ( int `  J ) `  ( X  \  S ) ) ) )  e.  J
)
133, 12eqeltrd 2512 1  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( X  \  (
( cls `  J
) `  S )
)  e.  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    \ cdif 3319    C_ wss 3322   U.cuni 4017   ` cfv 5456   Topctop 16960   intcnt 17083   clsccl 17084
This theorem is referenced by:  elcls  17139
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-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-top 16965  df-cld 17085  df-ntr 17086  df-cls 17087
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