# Basic Universe is Inductive

## Theorem

Let $V$ be a basic universe.

Then $V$ is an inductive class.

## Proof

By definition of basic universe, $V$ is a class containing all sets as elements.

By the Axiom of the Empty Set:

the empty class $\O$ is a set.

Hence $\O$ is an element of $V$.

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

By definition of power set:

$\set x \subseteq \powerset x$

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

By the Axiom of Unions, if $x$ is a set, then $\bigcup x$ is a set.

Thus $x \cup \set x$ is a set.

The result follows.

$\blacksquare$