Natural Number Ordering is Preserved by Successor Mapping

Theorem
Let $\N$ be the natural numbers.

Let $m, n \in \N$.

Then:
 * $n \le m \implies n^+ \le m^+$

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.

From Successor Mapping on Natural Numbers is Progressing, this successor mapping is a progressing mapping.

From Characteristics of Minimally Inductive Class under Progressing Mapping: Mapping Preserves Subsets:
 * $\forall m, n \in \omega: m \subseteq n \implies m^+ \subseteq n^+$