Theorem disjdifg

Description: A class does not intersect a relative complement of a superclass. (Contributed by NM, 24-Mar-1998) Generalize from disjdif . (Revised by BJ, 19-Jul-2026)

	Ref	Expression
Assertion	disjdifg	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∩ ( 𝐶 ∖ 𝐵 ) ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		ssinss1	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∩ 𝐶 ) ⊆ 𝐵 )
2		inssdif0	⊢ ( ( 𝐴 ∩ 𝐶 ) ⊆ 𝐵 ↔ ( 𝐴 ∩ ( 𝐶 ∖ 𝐵 ) ) = ∅ )
3	1 2	sylib	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∩ ( 𝐶 ∖ 𝐵 ) ) = ∅ )