Mapping from Set to Ordinal Class is Bounded Above

Theorem

Let $x$ be a set.

Let $\operatorname{On}$ be the class of all ordinals.

Let $f: x \to \operatorname{On}$ be a mapping.

Then $f$ has an upper bound.

Corollary

Let $\left\langle{x_n}\right\rangle$ be any ordinal-valued sequence.

Then $\left\langle{x_n}\right\rangle$ is bounded above.

Proof

This page is beyond the scope of ZFC, and should not be used in anything other than the theory in which it resides.

Let $I$ be the image of $f$.
By Union of Subset of Ordinals is Ordinal: Corollary, $\bigcup I$ is an ordinal.
But by Union is Smallest Superset, each element of $I$ is a subset of $\bigcup I$.
Thus $\bigcup I$ is an upper bound of $f$.
$\blacksquare$