Theorem disjdif

Description: A class and its relative complement are disjoint. Theorem 38 of Suppes p. 29. (Contributed by NM, 24-Mar-1998)

	Ref	Expression
Assertion	disjdif	⊢ ( 𝐴 ∩ ( 𝐵 ∖ 𝐴 ) ) = ∅

Proof

Step	Hyp	Ref	Expression
1		inss1	⊢ ( 𝐴 ∩ 𝐵 ) ⊆ 𝐴
2		inssdif0	⊢ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐴 ↔ ( 𝐴 ∩ ( 𝐵 ∖ 𝐴 ) ) = ∅ )
3	1 2	mpbi	⊢ ( 𝐴 ∩ ( 𝐵 ∖ 𝐴 ) ) = ∅