User:Dfeuer/Membership is Asymmetric on Natural Numbers

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

Then $\lnot(m \in n \land n \in m)$.

Proof
Suppose for the sake of contradiction that $m \in n$ and $n \in m$.

By User:Dfeuer/Natural Number is Transitive, $m$ is a transitive class.

Thus $m \in m$, contradicting User:Dfeuer/Natural Number does not Contain Itself.