Natural Numbers are Comparable/Strong Result

Theorem
Let $\N$ be the natural numbers.

Let $m, n \in \N$.

Then either:
 * $(1): \quad m + 1 \le n$

or:
 * $(2): \quad n \le m$

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.

Then from Minimally Inductive Class under Progressing Mapping induces Nest, $\omega$ is a nest in which:
 * $\forall m, n \in \omega: \map g x \subseteq y \lor y \subseteq x$

From the definition of $\map g x$ in this context:
 * $\forall x \in \omega: \map g x = x^+$

That is:


 * $\forall m, n \in \N: m + 1 \subseteq n \lor n \subseteq m$

whence the result.