Basic Universe is Supercomplete

Theorem

Let $V$ be a basic universe.

Then $V$ is supercomplete.

Proof

By definition, a class $V$ is supercomplete if and only if $V$ is both transitive and swelled.

From the Axiom of Transitivity, $V$ is transitive.

From the Axiom of Swelledness, $V$ is swelled.

Hence the result.

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 2$ Transitivity and supercompleteness