Cardinal of Finite Ordinal

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

Let $|n|$ denote the cardinal number of $n$.

Then it follows that:


 * $|n| = n$

Proof
Since $n$ is an ordinal, it follows that $|n| \le n$ by Cardinal Number Less than Ordinal: Corollary.

This means that $|n|$ is a member of the minimal infinite successor set.

Since $n$ is an ordinal, it also follows that $n \sim |n|$ by Ordinal Number Equivalent to Cardinal Number.

By Equality of Natural Numbers, it follows that $n = |n|$.