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Theorem ntrclsrcomplex 40573
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 40572 . 2 (𝜑𝐵 ∈ V)
4 difssd 4093 . 2 (𝜑 → (𝐵𝑆) ⊆ 𝐵)
53, 4sselpwd 5211 1 (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  Vcvv 3479  cdif 3915  𝒫 cpw 4520   class class class wbr 5047  cfv 6336
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-iota 6295  df-fv 6344
This theorem is referenced by:  ntrclsfveq1  40598  ntrclsfveq2  40599  ntrclsfveq  40600  ntrclsss  40601  ntrclsneine0lem  40602  ntrclsk2  40606  ntrclskb  40607  ntrclsk4  40610
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