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Theorem kur14lem4 30445
Description: Lemma for kur14 30452. 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 3820 . 2 (𝐴𝑋 ↔ (𝑋 ∖ (𝑋𝐴)) = 𝐴)
31, 2mpbi 219 1 (𝑋 ∖ (𝑋𝐴)) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1475  wcel 1977  cdif 3537  wss 3540   cuni 4372  cfv 5804  Topctop 20517  intcnt 20631  clsccl 20632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554
This theorem is referenced by:  kur14lem7  30448
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