Finite Ordinal is equal to Natural Number

Theorem
Let $n$ be a member of the minimal infinite successor set.

Let $x$ be an ordinal.

Then:


 * $n \sim x \implies n = x$

Proof
If $n \ne x$, then either $n < x$ or $x < n$ by Ordinal Membership Trichotomy.

If $x < n$, then both $x$ and $n$ are finite by Subset of Finite Set is Finite

Therefore, $x \not \sim n$ by Proper Subset of Finite Set No Bijection.

If $n < x$, then if $x$ is a finite set, then $x \not \sim n$ by Subset of Finite Set is Finite.

If $x$ is not a finite set, then if $x \sim n$, then $x$ is finite.

This would contradict the fact that $x$ is infinite.

In either case $x \not \sim n$.

Therefore, $n \sim x \implies n = x$ by contraposition.