Theorem kur14lem4 35634

Description: Lemma for kur14 35641. 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 4230	. 2 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴)
3	1, 2	mpbi 233	1 ⊢ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴

Syntax hints:   = wceq 1567   ∈ wcel 2149   ∖ cdif 3910   ⊆ wss 3913  ∪ cuni 4876  ‘cfv 6537  Topctop 23019  intcnt 23143  clsccl 23144

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-in 3920  df-ss 3930

This theorem is referenced by:  kur14lem7  35637