Basic Universe is not Set

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Let $V$ be a basic universe.

Then $V$ is not a set.


Aiming for a contradiction, suppose $V$ 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 $V$.

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.