Natural Numbers cannot be Elements of Each Other

Theorem
Let $m$ and $n$ be natural numbers.

Then it cannot be the case that both $m \in n$ and $n \in m$.

Proof
both $m \in n$ and $n \in m$.

We have $m \in n$

From Natural Number is Transitive Set:
 * $m \subseteq n$

by definition of transitive.

Thus:
 * $n \in m \subseteq n$

and so:
 * $n \in n$

But from Natural Number is Ordinary Set:
 * $n \notin n$

The result follows by Proof by Contradiction.