Theorem reldif

Description: A difference cutting down a relation is a relation. (Contributed by NM, 31-Mar-1998)

	Ref	Expression
Assertion	reldif	⊢ ( Rel 𝐴 → Rel ( 𝐴 ∖ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		difss	⊢ ( 𝐴 ∖ 𝐵 ) ⊆ 𝐴
2		relss	⊢ ( ( 𝐴 ∖ 𝐵 ) ⊆ 𝐴 → ( Rel 𝐴 → Rel ( 𝐴 ∖ 𝐵 ) ) )
3	1 2	ax-mp	⊢ ( Rel 𝐴 → Rel ( 𝐴 ∖ 𝐵 ) )