\begin{definition}[Definition:Cofinal Relation on Ordinals]
Let $x$ and $y$ be ordinals.
Then $y$ is said to be '''cofinal''' with respect to $x$ {{iff}} there exists a mapping $f: y \to x$ such that:
:$(1): \quad y \le x$
:$(2): \quad f$ is strictly increasing.
:$(3): \quad$ For all $a \in x$, there is some $b \in y$ such that $f \left({b}\right) \ge a$.
\end{definition}