Ordinal is Finite iff Natural Number

Theorem
Let $x$ be an ordinal.

Then, $x$ is a finite set iff $x$ is an element of the minimal infinite successor set.

Proof
$x$ is finite iff $x \sim \N_n$ for some $n \in \N$, by definition.

But $x$ is an ordinal, and by definition, it is equal to its initial segment.

By Natural Numbers are Elements of Minimal Infinite Successor Set, it follows that $x \sim n$ for some $n$.

By Ordinal Equivalent to Natural Number, it follows that $x$ is equal to $n$.

Thus, $x$ is an element of the minimal infinite successor set.