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Theorem clsneircomplex 44093
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 44092 . 2 (𝜑𝐵 ∈ V)
5 difssd 4147 . 2 (𝜑 → (𝐵𝑆) ⊆ 𝐵)
64, 5sselpwd 5334 1 (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  cdif 3960  𝒫 cpw 4605   class class class wbr 5148  ccom 5693  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fv 6571
This theorem is referenced by:  clsneiel2  44099
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