Intersection of Class of Sets is Set

Theorem
The intersection of a non-empty class $\Bbb C$ is a set.

Proof
Since $\Bbb C$ is a non-empty class, there is an $S \in \Bbb C$.

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

By definition of class intersection:
 * $x \in \ds \bigcap \Bbb C \implies x \in S$

By the subclass definition:
 * $\ds \bigcap \Bbb C \subseteq S$

By Subclass of Set is Set, $\ds \bigcap \Bbb C$ is a set.