Class is Subclass of Universal Class

Theorem
Let $V$ denote the universal class.

Let $A$ be a class.

Then $A$ is a subclass of $V$.

Proof
By definition of class, $A$ is a collection of sets.

Let $x \in A$ be a set.

By definition of universal class, $V$ contains all sets as elements.

Hence $x \in V$.

So we have that:
 * $x \in A \implies x \in V$

and the result follows by definition of subclass.