Ordinals under Multiplication form Ordered Monoid

Theorem
$\left({\operatorname{On}, \cdot, \le}\right)$ forms an ordered monoid, where:


 * $\operatorname{On}$ denotes the ordinal class, and
 * $\cdot$ denotes ordinal multiplication.

Proof
The result follows from Ordinals under Multiplication form Monoid and Ordinals under Multiplication form Ordered Semigroup.