Proceeding directly from the definitions, we have \begin{equation} \begin{split} B \setminus A & = \{ x : x \in B \text{ and } x \not\in A \} \\ & = \{ x : x \in B \subseteq X \text{ and } x \not\in A \} \\ & = \{ x : x \in B \text{ and } x \in X \text{ and } x \not\in A \} \\ & = \{ x : x \in B \text{ and } x \in X \setminus A \} \\ & = B \cap (X \setminus A) \end{split} \end{equation} $\square$