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Theorem kur14lem4 23740
Description: Lemma for kur14 23747. 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 3403 . 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 1623    e. wcel 1684    \ cdif 3149    C_ wss 3152   U.cuni 3827   ` cfv 5255   Topctop 16631   intcnt 16754   clsccl 16755
This theorem is referenced by:  kur14lem7  23743
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166
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