If $X \setminus Y$ is empty, then it is a subset of $X$ due to R7: Empty set is subset of every set. If $X \setminus Y$ is not empty, then $x$ being an element of $X \setminus Y$ implies that $x$ is an element of $X$ by the definition of set difference. Since this is true for every $x \in X \setminus Y$, the claim follows. $\square$