Existence of Hartogs Number/Proof 1
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Theorem
Let $S$ be a set.
Then $S$ has a Hartogs number.
Proof
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.
$\blacksquare$