Theorem difin0

Description: The difference of a class from its intersection is empty. Theorem 37 of Suppes p. 29. (Contributed by NM, 17-Aug-2004) (Proof shortened by Andrew Salmon, 26-Jun-2011)

	Ref	Expression
Assertion	difin0	$$⊢\left(A\cap B\right)\setminus B=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		inss2	$$⊢A\cap B\subseteq B$$
2		ssdif0	$$⊢A\cap B\subseteq B\leftrightarrow \left(A\cap B\right)\setminus B=\varnothing$$
3	1 2	mpbi	$$⊢\left(A\cap B\right)\setminus B=\varnothing$$