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Theorem neicvgrcomplex 38728
 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
neicvgbex.d 𝐷 = (𝑃𝐵)
neicvgbex.h 𝐻 = (𝐹 ∘ (𝐷𝐺))
neicvgbex.r (𝜑𝑁𝐻𝑀)
Assertion
Ref Expression
neicvgrcomplex (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)

Proof of Theorem neicvgrcomplex
StepHypRef Expression
1 neicvgbex.d . . 3 𝐷 = (𝑃𝐵)
2 neicvgbex.h . . 3 𝐻 = (𝐹 ∘ (𝐷𝐺))
3 neicvgbex.r . . 3 (𝜑𝑁𝐻𝑀)
41, 2, 3neicvgbex 38727 . 2 (𝜑𝐵 ∈ V)
5 difssd 3771 . 2 (𝜑 → (𝐵𝑆) ⊆ 𝐵)
64, 5sselpwd 4840 1 (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ∖ cdif 3604  𝒫 cpw 4191   class class class wbr 4685   ∘ ccom 5147  ‘cfv 5926 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-iota 5889  df-fv 5934 This theorem is referenced by:  neicvgel2  38735
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