Finite Ordinal is equal to Natural Number

Theorem
Let $n$ be an element of the minimal infinite successor set.

Let $x$ be an ordinal.

Then:


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

Proof
Let $n \ne x$.

Then either $n < x$ or $x < n$ by Ordinal Membership Trichotomy.

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

Therefore by Proper Subset of Finite Set No Bijection:
 * $x \not \sim n$

Suppose $n < x$.

Then $x \sim n$ implies that $x$ is finite by definition.

By Proper Subset of Finite Set No Bijection:
 * $x \not \sim n$

Therefore by contraposition:
 * $n \sim x \implies n = x$