Theorem kur14lem4 35743

Description: Lemma for kur14 35750. 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 4222	. 2 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴)
3	1, 2	mpbi 233	1 ⊢ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2146   ∖ cdif 3903   ⊆ wss 3906  ∪ cuni 4874  ‘cfv 6541  Topctop 23105  intcnt 23229  clsccl 23230

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-in 3913  df-ss 3923

This theorem is used by:  kur14lem7  35746