Natural Number m is Less than n implies n is not Greater than Successor of n/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 is then a direct application of Characteristics of Minimally Inductive Class under Progressing Mapping: Image of Proper Subset is Subset:
 * $m \subset n \implies m^+ \subseteq n$