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Theorem ntrclsrcomplex 37854
Description: The relative complement of the class 𝑆 exists as a subset of the base set. (Contributed by RP, 25-Jun-2021.)
Hypotheses
Ref Expression
ntrclsbex.d 𝐷 = (𝑂𝐵)
ntrclsbex.r (𝜑𝐼𝐷𝐾)
Assertion
Ref Expression
ntrclsrcomplex (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)

Proof of Theorem ntrclsrcomplex
StepHypRef Expression
1 ntrclsbex.d . . 3 𝐷 = (𝑂𝐵)
2 ntrclsbex.r . . 3 (𝜑𝐼𝐷𝐾)
31, 2ntrclsbex 37853 . 2 (𝜑𝐵 ∈ V)
4 difssd 3722 . 2 (𝜑 → (𝐵𝑆) ⊆ 𝐵)
53, 4sselpwd 4777 1 (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  Vcvv 3190  cdif 3557  𝒫 cpw 4136   class class class wbr 4623  cfv 5857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-iota 5820  df-fv 5865
This theorem is referenced by:  ntrclsfveq1  37879  ntrclsfveq2  37880  ntrclsfveq  37881  ntrclsss  37882  ntrclsneine0lem  37883  ntrclsk2  37887  ntrclskb  37888  ntrclsk4  37891
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