Intersection of Non-Empty Class is Set/Proof 1

Proof
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 the, $V$ is a swelled class.

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

It follows $\bigcap A$ is a set.