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Theorem kur14lem4 24896
Description: Lemma for kur14 24903. Complementation is an involution on the set of subsets of a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
Hypotheses
Ref Expression
kur14lem.j  |-  J  e. 
Top
kur14lem.x  |-  X  = 
U. J
kur14lem.k  |-  K  =  ( cls `  J
)
kur14lem.i  |-  I  =  ( int `  J
)
kur14lem.a  |-  A  C_  X
Assertion
Ref Expression
kur14lem4  |-  ( X 
\  ( X  \  A ) )  =  A

Proof of Theorem kur14lem4
StepHypRef Expression
1 kur14lem.a . 2  |-  A  C_  X
2 dfss4 3576 . 2  |-  ( A 
C_  X  <->  ( X  \  ( X  \  A
) )  =  A )
31, 2mpbi 201 1  |-  ( X 
\  ( X  \  A ) )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726    \ cdif 3318    C_ wss 3321   U.cuni 4016   ` cfv 5455   Topctop 16959   intcnt 17082   clsccl 17083
This theorem is referenced by:  kur14lem7  24899
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-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-v 2959  df-dif 3324  df-in 3328  df-ss 3335
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