Ordinal is Less than Ordinal to Limit Power

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Theorem

Let $x$, $y$, and $z$ be ordinals.

Let $z$ be a limit ordinal.


Then:

$x < y^z \iff \exists w \in z: x < y^w$


Proof

\(\ds \) \(\) \(\ds x < y^z\)
\(\ds \leadstoandfrom \ \ \) \(\ds \) \(\) \(\ds x < \bigcup_{w \mathop \in z} y^w\) Definition of Ordinal Exponentiation
\(\ds \leadstoandfrom \ \ \) \(\ds \) \(\) \(\ds \exists w \in z: x < y^w\) Definition of Union of Family

$\blacksquare$


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