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Theorem kur14lem4 23755
Description: Lemma for kur14 23762. 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 3416 . 2  |-  ( A 
C_  X  <->  ( X  \  ( X  \  A
) )  =  A )
31, 2mpbi 199 1  |-  ( X 
\  ( X  \  A ) )  =  A
Colors of variables: wff set class
Syntax hints:    = 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:  kur14lem7  23758
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-dif 3168  df-in 3172  df-ss 3179
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