Basic Universe is not Set

Theorem
Let $V$ be a basic universe.

Then $V$ is not a set.

Proof
$V$ were a set.

Then by the axiom of swelledness, $V$ is swelled.

That is, as $V$ is a set, every subclass of $V$ would also be a set.

From Class has Subclass which is not Element, $V$ has a subclass $S$ which is not an element of $V$.

That is:
 * $\exists S \subseteq V: S \notin V$

But by definition of a basic universe, $V$ is a universal class.

That is:
 * $S \in V$

This contradicts the deduction that $S \notin V$.

Hence the result by Proof by Contradiction.