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 ∈ 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 3858	. 2 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴)
3	1, 2	mpbi 220	1 ⊢ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴

Syntax hints:   = wceq 1483   ∈ wcel 1990   ∖ cdif 3571   ⊆ wss 3574  ∪ 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