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Theorem ntrclsrcomplex 38859
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 38858 . 2 (𝜑𝐵 ∈ V)
4 difssd 3889 . 2 (𝜑 → (𝐵𝑆) ⊆ 𝐵)
53, 4sselpwd 4941 1 (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  Vcvv 3351  cdif 3720  𝒫 cpw 4297   class class class wbr 4786  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039
This theorem is referenced by:  ntrclsfveq1  38884  ntrclsfveq2  38885  ntrclsfveq  38886  ntrclsss  38887  ntrclsneine0lem  38888  ntrclsk2  38892  ntrclskb  38893  ntrclsk4  38896
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