Intersection of Non-Empty Class is Set/Proof 2

Proof
Since $A$ is a non-empty class, there exists $S \in A$.

Since $S$ is an element of a class, it is not a proper class, and is thus a set.

By definition of class intersection:
 * $x \in \bigcap A \implies x \in S$

By the subclass definition:
 * $\bigcap A \subseteq S$

By Subclass of Set is Set, $\bigcap A$ is a set.