Element of Finite Ordinal iff Subset

Theorem
Let $\N$ be the natural numbers, defined as the minimal infinite successor set $\omega$.

Let $m, n \in \N$ such that $m \ne n$.

Then:
 * $m \in n \iff m \subseteq n$

Proof
Let $m \in n$.

From Natural Numbers are Transitive Sets it follows directly that $m \subseteq n$.

Now let $m \subseteq n$.

We have by hypothesis that $m \ne n$.

From Natural Numbers are Comparable‎ it follows that either $m \in n$ or $n \in m$.

Suppose $n \in m$.

Then $\exists x \in m: m \subseteq x$ which contradicts Natural Number is Not Subset of Element.

The only option left is that $m \in n$.

Hence the result.