Theorem kur14lem4 34017

Description: Lemma for kur14 34024. 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 ∈ Top
kur14lem.x	⊢ 𝑋 = ∪ 𝐽
kur14lem.k	⊢ 𝐾 = (cls‘𝐽)
kur14lem.i	⊢ 𝐼 = (int‘𝐽)
kur14lem.a	⊢ 𝐴 ⊆ 𝑋

Assertion

Ref	Expression
kur14lem4	⊢ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴

Proof of Theorem kur14lem4

Step	Hyp	Ref	Expression
1		kur14lem.a	. 2 ⊢ 𝐴 ⊆ 𝑋
2		dfss4 4253	. 2 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴)
3	1, 2	mpbi 229	1 ⊢ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴

Syntax hints:   = wceq 1541   ∈ wcel 2106   ∖ cdif 3940   ⊆ wss 3943  ∪ cuni 4900  ‘cfv 6531  Topctop 22321  intcnt 22447  clsccl 22448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3474  df-dif 3946  df-in 3950  df-ss 3960

This theorem is referenced by:  kur14lem7  34020