Existence of Hartogs Number/Proof 1

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Let $S$ be a set.

Then $S$ has a Hartogs number.


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.