Theorem neicvgrcomplex 44557

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 = (𝑃‘𝐵)
neicvgbex.h	⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))
neicvgbex.r	⊢ (𝜑 → 𝑁𝐻𝑀)

Assertion

Ref	Expression
neicvgrcomplex	⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)

Proof of Theorem neicvgrcomplex

Step	Hyp	Ref	Expression
1		neicvgbex.d	. . 3 ⊢ 𝐷 = (𝑃‘𝐵)
2		neicvgbex.h	. . 3 ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))
3		neicvgbex.r	. . 3 ⊢ (𝜑 → 𝑁𝐻𝑀)
4	1, 2, 3	neicvgbex 44556	. 2 ⊢ (𝜑 → 𝐵 ∈ V)
5		difssd 4067	. 2 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ⊆ 𝐵)
6	4, 5	sselpwd 5256	1 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  Vcvv 3431   ∖ cdif 3880  𝒫 cpw 4529   class class class wbr 5072   ∘ ccom 5622  ‘cfv 6485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-iota 6441  df-fv 6493

This theorem is referenced by:  neicvgel2  44564