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Theorem cmclsopn 19671
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 19659 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  ( X  \ 
( ( int `  J
) `  ( X  \  S ) ) ) )
32difeq2d 3553 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( X  \  (
( cls `  J
) `  S )
)  =  ( X 
\  ( X  \ 
( ( int `  J
) `  ( X  \  S ) ) ) ) )
4 difss 3562 . . . . . . 7  |-  ( X 
\  S )  C_  X
51ntropn 19658 . . . . . . 7  |-  ( ( J  e.  Top  /\  ( X  \  S ) 
C_  X )  -> 
( ( int `  J
) `  ( X  \  S ) )  e.  J )
64, 5mpan2 669 . . . . . 6  |-  ( J  e.  Top  ->  (
( int `  J
) `  ( X  \  S ) )  e.  J )
71eltopss 19524 . . . . . 6  |-  ( ( J  e.  Top  /\  ( ( int `  J
) `  ( X  \  S ) )  e.  J )  ->  (
( int `  J
) `  ( X  \  S ) )  C_  X )
86, 7mpdan 666 . . . . 5  |-  ( J  e.  Top  ->  (
( int `  J
) `  ( X  \  S ) )  C_  X )
9 dfss4 3674 . . . . 5  |-  ( ( ( int `  J
) `  ( X  \  S ) )  C_  X 
<->  ( X  \  ( X  \  ( ( int `  J ) `  ( X  \  S ) ) ) )  =  ( ( int `  J
) `  ( X  \  S ) ) )
108, 9sylib 196 . . . 4  |-  ( J  e.  Top  ->  ( X  \  ( X  \ 
( ( int `  J
) `  ( X  \  S ) ) ) )  =  ( ( int `  J ) `
 ( X  \  S ) ) )
1110, 6eqeltrd 2484 . . 3  |-  ( J  e.  Top  ->  ( X  \  ( X  \ 
( ( int `  J
) `  ( X  \  S ) ) ) )  e.  J )
1211adantr 463 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( X  \  ( X  \  ( ( int `  J ) `  ( X  \  S ) ) ) )  e.  J
)
133, 12eqeltrd 2484 1  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( X  \  (
( cls `  J
) `  S )
)  e.  J )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1836    \ cdif 3403    C_ wss 3406   U.cuni 4180   ` cfv 5513   Topctop 19502   intcnt 19626   clsccl 19627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-reu 2753  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-int 4217  df-iun 4262  df-iin 4263  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-top 19507  df-cld 19628  df-ntr 19629  df-cls 19630
This theorem is referenced by:  elcls  19683
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