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Theorem kur14lem4 35180
Description: Lemma for kur14 35187. 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 4288 . 2 (𝐴𝑋 ↔ (𝑋 ∖ (𝑋𝐴)) = 𝐴)
31, 2mpbi 230 1 (𝑋 ∖ (𝑋𝐴)) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2103  cdif 3973  wss 3976   cuni 4937  cfv 6579  Topctop 22924  intcnt 23050  clsccl 23051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3440  df-v 3486  df-dif 3979  df-in 3983  df-ss 3993
This theorem is referenced by:  kur14lem7  35183
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