Critique my proof of: Suppose $A$, $B$, and $C$ are sets, and $A\setminus B \subseteq C$. Then $A\setminus C \subseteq B$.

I'd avoid the usage of so many symbols.

Your proof is too long. Suppose $x\in A\setminus C$. Then $x\in A$ and $x\notin C$. Suppose, for the sake of contradiction, that $x\notin B$; thus, by definition, we get that $x\in A\setminus B$ and therefore, by assumption, $x\in C$, which is a contradiction. Thus $x\in B$.

Your proof is good except for the statement 'Therefore, $x \in A \land x \notin C \Rightarrow x \in B$.' Let me correct it to

The goal is to show that $\forall x:[x \in A \land x \notin C \Rightarrow x \in B]$. Let $x \in A \setminus C$ be arbitrary.

Put the remaining part of your proof after this.