Class is Subclass of Universal Class

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Let $V$ denote the universal class.

Let $A$ be a class.

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


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.