Ordinal Number Equivalent to Cardinal Number

Theorem

Let $x$ be an ordinal.

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

Then:

$x \sim \card x$

where $\sim$ denotes set equivalence.

Proof

$x \sim x$

Therefore, $x$ is equivalent to some ordinal.

$x \sim \card x$

$\blacksquare$