Trichotomy Law for Natural Numbers

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

Let $m, n \in \omega$.

Then one of the following cases holds:
 * $m \in n$
 * $m = n$
 * $n \in m$

Proof
By definition of the ordering on von Neumann construction:


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

From Natural Number m is Less than n iff m is an Element of n, we have:


 * $m < n \iff m \in n$

Hence the theorem is equivalent to the statement that for every $m, n \in \omega$, one of the following holds:
 * $m \subsetneq n$
 * $m = n$
 * $n \subsetneq m$

The result follows from Natural Numbers are Comparable: Proof using Von Neumann Construction.