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Theorem kur14lem4 34031
Description: Lemma for kur14 34038. 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 = βˆͺ 𝐽
kur14lem.k 𝐾 = (clsβ€˜π½)
kur14lem.i 𝐼 = (intβ€˜π½)
kur14lem.a 𝐴 βŠ† 𝑋
Assertion
Ref Expression
kur14lem4 (𝑋 βˆ– (𝑋 βˆ– 𝐴)) = 𝐴

Proof of Theorem kur14lem4
StepHypRef Expression
1 kur14lem.a . 2 𝐴 βŠ† 𝑋
2 dfss4 4254 . 2 (𝐴 βŠ† 𝑋 ↔ (𝑋 βˆ– (𝑋 βˆ– 𝐴)) = 𝐴)
31, 2mpbi 229 1 (𝑋 βˆ– (𝑋 βˆ– 𝐴)) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541   ∈ wcel 2106   βˆ– cdif 3941   βŠ† wss 3944  βˆͺ cuni 4901  β€˜cfv 6532  Topctop 22324  intcnt 22450  clsccl 22451
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3947  df-in 3951  df-ss 3961
This theorem is referenced by:  kur14lem7  34034
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