Existence of Hartogs Number/Proof 2

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Theorem

Let $S$ be a set.


Then $S$ has a Hartogs number.


Proof

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

The collection of all dominated ordinals:

$\set {y \in \On: y \preccurlyeq S}$

is the Hartogs number of $S$.

$\blacksquare$