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Theorem kur14lem4 28882
Description: Lemma for kur14 28889. 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 3674 . 2  |-  ( A 
C_  X  <->  ( X  \  ( X  \  A
) )  =  A )
31, 2mpbi 208 1  |-  ( X 
\  ( X  \  A ) )  =  A
Colors of variables: wff setvar class
Syntax hints:    = 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-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-v 3053  df-dif 3409  df-in 3413  df-ss 3420
This theorem is referenced by:  kur14lem7  28885
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