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Theorem kur14lem4 31798
 Description: Lemma for kur14 31805. 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 4085 . 2 (𝐴𝑋 ↔ (𝑋 ∖ (𝑋𝐴)) = 𝐴)
31, 2mpbi 222 1 (𝑋 ∖ (𝑋𝐴)) = 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1601   ∈ wcel 2107   ∖ cdif 3789   ⊆ wss 3792  ∪ cuni 4673  ‘cfv 6137  Topctop 21116  intcnt 21240  clsccl 21241 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-v 3400  df-dif 3795  df-in 3799  df-ss 3806 This theorem is referenced by:  kur14lem7  31801
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