Existence of Hartogs Number

Theorem
Let $S$ be a set.

Then $S$ has a Hartogs number.

Proof 1
From Hartogs' lemma there exists an ordinal $\alpha$ such that there is no injection from $\alpha$ to $S$.

We also have that Ordinals are Well-Ordered.

It follows from the definition of well-ordering that there exists a smallest such ordinal.

Hence the result.

Proof 2
Follows immediately from Cardinal Equal to Collection of All Dominated Ordinals.

The collection of all dominated ordinals, $\left\{ y \in \operatorname{On} : y \preccurlyeq S \right\}$ is the Hartogs number of $S$.