Prove that if $A\subseteq B$ then $A\setminus C\subseteq B\setminus C$ How can I prove that if $$A \subseteq B$$ then $$A\setminus C \subseteq B \setminus C$$? 
I'm a bit lost. I have an idea that follows like so:
Let $x \in A, x \notin C$, then $\{x\} \in P(A\setminus C); \{x\} \in P(B\setminus C)$, while saying that nullsets are subsets of nullsets if $x \in C$.
However, I believe that there is a more straightforward solution. How should I prove the statement above?

Hint: Consider two cases: 
Case 1:  $A\subseteq C$ and in this case $A\setminus C=\emptyset$. What can you conclude?
Case 2:  $A\setminus C\neq\emptyset$ and choose $x\in A\setminus C$. What can you conclude?
