Suppose $A$, $B$, and $C$ are sets, and $A - B \subseteq C$. Then $A - C \subseteq B$. I know how to prove it by contradiction, but I am wondering if it's possible to prove it directly. I tried doing that, but so far no results. Is it not possible to prove it directly?
Thanks.

We have
$$A-B=A\cap B^c\subset C\Rightarrow C^c\subset A^c\cup B$$
hence
$$A-C=A\cap C^c\subset A\cap  (A^c\cup B)=A\cap B\subset B$$
