Theorem resnonrel

Description: A restriction of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020)

	Ref	Expression
Assertion	resnonrel	⊢ ( ( 𝐴 ∖ ◡ ◡ 𝐴 ) ↾ 𝐵 ) = ∅

Proof

Step	Hyp	Ref	Expression
1		ssv	⊢ 𝐵 ⊆ V
2		ssres2	⊢ ( 𝐵 ⊆ V → ( ( 𝐴 ∖ ◡ ◡ 𝐴 ) ↾ 𝐵 ) ⊆ ( ( 𝐴 ∖ ◡ ◡ 𝐴 ) ↾ V ) )
3	1 2	ax-mp	⊢ ( ( 𝐴 ∖ ◡ ◡ 𝐴 ) ↾ 𝐵 ) ⊆ ( ( 𝐴 ∖ ◡ ◡ 𝐴 ) ↾ V )
4		cnvnonrel	⊢ ◡ ( 𝐴 ∖ ◡ ◡ 𝐴 ) = ∅
5	4	cnveqi	⊢ ◡ ◡ ( 𝐴 ∖ ◡ ◡ 𝐴 ) = ◡ ∅
6		cnvcnv2	⊢ ◡ ◡ ( 𝐴 ∖ ◡ ◡ 𝐴 ) = ( ( 𝐴 ∖ ◡ ◡ 𝐴 ) ↾ V )
7		cnv0	⊢ ◡ ∅ = ∅
8	5 6 7	3eqtr3i	⊢ ( ( 𝐴 ∖ ◡ ◡ 𝐴 ) ↾ V ) = ∅
9	3 8	sseqtri	⊢ ( ( 𝐴 ∖ ◡ ◡ 𝐴 ) ↾ 𝐵 ) ⊆ ∅
10		ss0b	⊢ ( ( ( 𝐴 ∖ ◡ ◡ 𝐴 ) ↾ 𝐵 ) ⊆ ∅ ↔ ( ( 𝐴 ∖ ◡ ◡ 𝐴 ) ↾ 𝐵 ) = ∅ )
11	9 10	mpbi	⊢ ( ( 𝐴 ∖ ◡ ◡ 𝐴 ) ↾ 𝐵 ) = ∅