Critique my proof of: Suppose A, B, and C are sets. Then $A \cap (B \setminus C) = (A \cap B) \setminus C$.
Here is another way to approach it for the sake of curiosity.
According to the properties and definitions involved, we can conclude that
\begin{align*} A\cap(B\backslash C) & = A\cap (B\cap C^{c})\\\\ & = (A\cap B)\cap C^{c}\\\\ & = (A\cap B)\backslash C \end{align*}
and we are done.
As to your proof, it sounds good.
Hopefully this helps!