Theorem ntrclselnel2

Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is an equivalence between membership in interior of the complement of a set and non-membership in the closure of the set. (Contributed by RP, 28-May-2021)

	Ref	Expression
Hypotheses	ntrcls.o	⊢ 𝑂 = ( 𝑖 ∈ V ↦ ( 𝑘 ∈ ( 𝒫 𝑖 ↑m 𝒫 𝑖 ) ↦ ( 𝑗 ∈ 𝒫 𝑖 ↦ ( 𝑖 ∖ ( 𝑘 ‘ ( 𝑖 ∖ 𝑗 ) ) ) ) ) )
	ntrcls.d	⊢ 𝐷 = ( 𝑂 ‘ 𝐵 )
	ntrcls.r	⊢ ( 𝜑 → 𝐼 𝐷 𝐾 )
	ntrcls.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	ntrcls.s	⊢ ( 𝜑 → 𝑆 ∈ 𝒫 𝐵 )
Assertion	ntrclselnel2	⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝐼 ‘ ( 𝐵 ∖ 𝑆 ) ) ↔ ¬ 𝑋 ∈ ( 𝐾 ‘ 𝑆 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ntrcls.o	⊢ 𝑂 = ( 𝑖 ∈ V ↦ ( 𝑘 ∈ ( 𝒫 𝑖 ↑m 𝒫 𝑖 ) ↦ ( 𝑗 ∈ 𝒫 𝑖 ↦ ( 𝑖 ∖ ( 𝑘 ‘ ( 𝑖 ∖ 𝑗 ) ) ) ) ) )
2		ntrcls.d	⊢ 𝐷 = ( 𝑂 ‘ 𝐵 )
3		ntrcls.r	⊢ ( 𝜑 → 𝐼 𝐷 𝐾 )
4		ntrcls.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5		ntrcls.s	⊢ ( 𝜑 → 𝑆 ∈ 𝒫 𝐵 )
6	1 2 3	ntrclsnvobr	⊢ ( 𝜑 → 𝐾 𝐷 𝐼 )
7	1 2 6 4 5	ntrclselnel1	⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝐾 ‘ 𝑆 ) ↔ ¬ 𝑋 ∈ ( 𝐼 ‘ ( 𝐵 ∖ 𝑆 ) ) ) )
8	7	con2bid	⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝐼 ‘ ( 𝐵 ∖ 𝑆 ) ) ↔ ¬ 𝑋 ∈ ( 𝐾 ‘ 𝑆 ) ) )