Ordinal Membership is Asymmetric

Theorem

Let $m$ and $n$ be ordinals.

Then it is not the case that $m \in n$ and $n \in m$.

Proof

Aiming for a contradiction, suppose $m \in n$ and $n \in m$.

Since $m$ is an ordinal, it is transitive.

Thus since $m \in n$ and $n \in m$, it follows that $m \in m$.

But this contradicts Ordinal is not Element of Itself.

$\blacksquare$