Theorem neicvgrcomplex

Description: The relative complement of the class S exists as a subset of the base set. (Contributed by RP, 26-Jun-2021)

	Ref	Expression
Hypotheses	neicvgbex.d	⊢ 𝐷 = ( 𝑃 ‘ 𝐵 )
	neicvgbex.h	⊢ 𝐻 = ( 𝐹 ∘ ( 𝐷 ∘ 𝐺 ) )
	neicvgbex.r	⊢ ( 𝜑 → 𝑁 𝐻 𝑀 )
Assertion	neicvgrcomplex	⊢ ( 𝜑 → ( 𝐵 ∖ 𝑆 ) ∈ 𝒫 𝐵 )

Proof

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