Theorem kur14lem4 31191

Description: Lemma for kur14 31198. 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

Step	Hyp	Ref	Expression
1		kur14lem.a	. 2 |- A C_ X
2		dfss4 3858	. 2 |- ( A C_ X <-> ( X \ ( X \ A ) ) = A )
3	1, 2	mpbi 220	1 |- ( X \ ( X \ A ) ) = A

Syntax hints:   = wceq 1483   e. wcel 1990   \ cdif 3571   C_ wss 3574   U.cuni 4436   ` cfv 5888   Topctop 20698   intcnt 20821   clsccl 20822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588

This theorem is referenced by:  kur14lem7  31194