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Theorem kur14lem4 32581
 Description: Lemma for kur14 32588. 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 𝐴𝑋
Assertion
Ref Expression
kur14lem4 (𝑋 ∖ (𝑋𝐴)) = 𝐴

Proof of Theorem kur14lem4
StepHypRef Expression
1 kur14lem.a . 2 𝐴𝑋
2 dfss4 4185 . 2 (𝐴𝑋 ↔ (𝑋 ∖ (𝑋𝐴)) = 𝐴)
31, 2mpbi 233 1 (𝑋 ∖ (𝑋𝐴)) = 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2111   ∖ cdif 3878   ⊆ wss 3881  ∪ cuni 4800  ‘cfv 6324  Topctop 21505  intcnt 21629  clsccl 21630 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-rab 3115  df-v 3443  df-dif 3884  df-in 3888  df-ss 3898 This theorem is referenced by:  kur14lem7  32584
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