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

Proof

Step	Hyp	Ref	Expression
1		inss1	$$⊢A\cap B\subseteq A$$
2		inssdif0	$$⊢A\cap B\subseteq A\leftrightarrow A\cap \left(B\setminus A\right)=\varnothing$$
3	1 2	mpbi	$$⊢A\cap \left(B\setminus A\right)=\varnothing$$