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Theorem ntrclsrcomplex 39119
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 39118 . 2 (𝜑𝐵 ∈ V)
4 difssd 3940 . 2 (𝜑 → (𝐵𝑆) ⊆ 𝐵)
53, 4sselpwd 5006 1 (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  Vcvv 3389  cdif 3770  𝒫 cpw 4353   class class class wbr 4847  cfv 6105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2379  ax-ext 2781  ax-sep 4979  ax-nul 4987  ax-pow 5039
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2593  df-eu 2611  df-clab 2790  df-cleq 2796  df-clel 2799  df-nfc 2934  df-ne 2976  df-ral 3098  df-rex 3099  df-rab 3102  df-v 3391  df-dif 3776  df-un 3778  df-in 3780  df-ss 3787  df-nul 4120  df-if 4282  df-pw 4355  df-sn 4373  df-pr 4375  df-op 4379  df-uni 4633  df-br 4848  df-iota 6068  df-fv 6113
This theorem is referenced by:  ntrclsfveq1  39144  ntrclsfveq2  39145  ntrclsfveq  39146  ntrclsss  39147  ntrclsneine0lem  39148  ntrclsk2  39152  ntrclskb  39153  ntrclsk4  39156
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