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 |
$\blacksquare$
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 8.40$