Existence of Disjoint Well-Ordered Sets Isomorphic to Ordinals

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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.


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

\(\ds a\) \(:=\) \(\ds \alpha \times \set 0\)
\(\ds b\) \(:=\) \(\ds \beta \times \set 1\)

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

The result follows.