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 $S$.
$\blacksquare$