Von Neumann Construction of Natural Numbers is Minimally Inductive

Theorem
Let $\omega$ denote the set of natural numbers as defined by the von Neumann construction.

$\omega$ is a minimally inductive class under the successor mapping.

Proof
Consider Peano's axioms:

From Inductive Construction of Natural Numbers fulfils Peano's Axioms, $\omega$ fulfils Peano's axioms.

We note that from :
 * $\O \in \omega$

We acknowledge from :
 * the successor mapping defines that $n^+ := n \cup \set n$

and from the result follows.