No Natural Number between Number and Successor/Proof using Von Neumann Construction

Proof
Let $\N$ be defined as the von Neumann construction $\omega$.

By definition of the ordering on von Neumann construction:


 * $m \le n \iff m \subseteq n$

From Von Neumann Construction of Natural Numbers is Minimally Inductive, $\omega$ is minimally inductive class under the successor mapping.

The result follows from the Sandwich Principle:
 * $\forall m, n \in \omega: m \subseteq n \subseteq m^+ \implies m = n \lor n = m^+$

where $m^+$ is identified as $m + 1$.