Cardinal of Finite Ordinal

Theorem
Let $n$ be a finite ordinal.

Let $\card n$ denote the cardinal number of $n$.

Then:


 * $\card n = n$

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

Hence, $\card n$ is also a finite ordinal.

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

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