If A,B,C are sets such that $A\cap C\subseteq B$, if $a\in C\Rightarrow a\notin A\setminus B$

Question

I have to show : If A,B,C are sets such that $A\cap C\subseteq B$, if $a\in C\Rightarrow a\notin A\setminus B$

proof: suppose $a\in A\setminus B\Rightarrow a\in A$ and $a\notin B\rightarrow a\notin A\cap C$ since $A\cap C\subseteq B$, but this is a contracdiction since $a\in C$, am I correct?

Answer 1

I think your reasoning is fine (just make sure to write it in a more readable fashion).

Just make sure to point out that the contradiction is not only due to the fact that $a\in C$, but to $a\in A$ as well (since we need $a\in A\cap C$), which is exactly the hypothesis you're starting with.