Existence of Disjoint Well-Ordered Sets Isomorphic to Ordinals

Theorem
Let $\alpha$ and $\beta$ be ordinals.

Then there exist two well-ordered sets $a$ and $b$ such that:
 * $a$ and $b$ are order isomorphic to $\alpha$ and $\beta$ respectively
 * $a$ and $b$ are disjoint.

Proof
Let $a$ and $b$ be defined as:

Then let $a$ and $b$ be ordered by their first coordinate.

The result follows.