Theorem disj2

Description: Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998)

	Ref	Expression
Assertion	disj2	⊢ ( ( 𝐴 ∩ 𝐵 ) = ∅ ↔ 𝐴 ⊆ ( V ∖ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ssv	⊢ 𝐴 ⊆ V
2		reldisj	⊢ ( 𝐴 ⊆ V → ( ( 𝐴 ∩ 𝐵 ) = ∅ ↔ 𝐴 ⊆ ( V ∖ 𝐵 ) ) )
3	1 2	ax-mp	⊢ ( ( 𝐴 ∩ 𝐵 ) = ∅ ↔ 𝐴 ⊆ ( V ∖ 𝐵 ) )