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Theorem neicvgrcomplex 42053
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 42052 . 2 (𝜑𝐵 ∈ V)
5 difssd 4079 . 2 (𝜑 → (𝐵𝑆) ⊆ 𝐵)
64, 5sselpwd 5270 1 (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  Vcvv 3441  cdif 3895  𝒫 cpw 4547   class class class wbr 5092  ccom 5624  cfv 6479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-iota 6431  df-fv 6487
This theorem is referenced by:  neicvgel2  42060
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