Intersection of Non-Empty Class is Set

Theorem

Let $A$ be a non-empty class.

Let $\bigcap A$ denote the intersection of $A$.

Then $\bigcap A$ is a set.

Let $x$ be a non-empty set.

Let $\bigcap x$ denote the intersection of $x$.

Then $\bigcap x$ is a set.

Let $V$ denote the basic universe such that $A \subseteq V$.

We are given that $A$ is non-empty.

Then $\exists x \in A$ where $x$ is a set.

By definition of intersection of class, every element of $\bigcap A$ is an element of all elements of $A$.

Thus:

$\bigcap A \subseteq x$

We are given that $A$ is a subclass of the basic universe $V$.

Thus $x \in V$ by definition of basic universe.

By definition of swelled class, every subclass of a set $x \in V$ is a set.

It follows $\bigcap A$ is a set.

$\blacksquare$

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.

$x \in \bigcap A \implies x \in S$

By the subclass definition:

$\bigcap A \subseteq S$

$\blacksquare$