Bounded Class is Set

Theorem

Let $B$ be a class.

Let it be assumed that $B$ is a subclass of a basic universe $V$.

Let $B$ be bounded by a set $x$.

Then $B$ is itself a set.

By definition, every element of $B$ is a subset of $x$.

Then every element of $B$ is an element of the power set $\powerset x$ of $x$.

Thus $B$ is a subclass of $\powerset x$.

By the Axiom of Powers, $\powerset x$ is a set.

That is, $\powerset x$ is an element of $V$.

As $V$ is a swelled class, then by definition, then every subclass of $\powerset x$ is an element of $V$.

That includes $B$.

That is, $B$ is an element of $V$.

Thus, by definition of $V$, $B$ is a set.

$\blacksquare$

2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 4$ A double induction principle and its applications