Theorem ssdifin0

Description: A subset of a difference does not intersect the subtrahend. (Contributed by Jeff Hankins, 1-Sep-2013) (Proof shortened by Mario Carneiro, 24-Aug-2015)

	Ref	Expression
Assertion	ssdifin0	$$⊢A\subseteq B\setminus C\to A\cap C=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		ssrin	$$⊢A\subseteq B\setminus C\to A\cap C\subseteq \left(B\setminus C\right)\cap C$$
2		disjdifr	$$⊢\left(B\setminus C\right)\cap C=\varnothing$$
3		sseq0	$$⊢\left(A\cap C\subseteq \left(B\setminus C\right)\cap C\wedge \left(B\setminus C\right)\cap C=\varnothing \right)\to A\cap C=\varnothing$$
4	1 2 3	sylancl	$$⊢A\subseteq B\setminus C\to A\cap C=\varnothing$$