Theorem clsneircomplex

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	clsneibex.d	⊢ 𝐷 = ( 𝑃 ‘ 𝐵 )
	clsneibex.h	⊢ 𝐻 = ( 𝐹 ∘ 𝐷 )
	clsneibex.r	⊢ ( 𝜑 → 𝐾 𝐻 𝑁 )
Assertion	clsneircomplex	⊢ ( 𝜑 → ( 𝐵 ∖ 𝑆 ) ∈ 𝒫 𝐵 )

Proof

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