Intersection of Non-Empty Class is Set/Corollary

Theorem
Let $x$ be a non-empty set.

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

Then $\bigcap x$ is a set.

Proof
It is assumed that $x$ is an element of a basic universe.

Hence from the axiom of transitivity, every set is a class.

Hence Intersection of Non-Empty Class is Set applies directly.