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	$$⊢A\subseteq B\to A\cap \left(C\setminus B\right)=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		ssinss1	$$⊢A\subseteq B\to A\cap C\subseteq B$$
2		inssdif0	$$⊢A\cap C\subseteq B\leftrightarrow A\cap \left(C\setminus B\right)=\varnothing$$
3	1 2	sylib	$$⊢A\subseteq B\to A\cap \left(C\setminus B\right)=\varnothing$$