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Theorem clsneircomplex 44116
Description: The relative complement of the class 𝑆 exists as a subset of the base set. (Contributed by RP, 26-Jun-2021.)
Hypotheses
Ref Expression
clsneibex.d 𝐷 = (𝑃𝐵)
clsneibex.h 𝐻 = (𝐹𝐷)
clsneibex.r (𝜑𝐾𝐻𝑁)
Assertion
Ref Expression
clsneircomplex (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)

Proof of Theorem clsneircomplex
StepHypRef Expression
1 clsneibex.d . . 3 𝐷 = (𝑃𝐵)
2 clsneibex.h . . 3 𝐻 = (𝐹𝐷)
3 clsneibex.r . . 3 (𝜑𝐾𝐻𝑁)
41, 2, 3clsneibex 44115 . 2 (𝜑𝐵 ∈ V)
5 difssd 4137 . 2 (𝜑 → (𝐵𝑆) ⊆ 𝐵)
64, 5sselpwd 5328 1 (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  cdif 3948  𝒫 cpw 4600   class class class wbr 5143  ccom 5689  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fv 6569
This theorem is referenced by:  clsneiel2  44122
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