Subset is Right Compatible with Ordinal Exponentiation

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

Then:


 * $x \le y \implies x^z \le y^z$

Proof
The proof shall proceed by Transfinite Induction on $z$.

Basis for the Induction
If $z = \varnothing$, then $x^z = 1$

This proves the basis for the induction.

Induction Step
The inductive hypothesis states that $x^z \le y^z$ for $y$.

Then:

This proves the induction step.

Limit Case
The inductive hypothesis for the limit case states that:


 * $\forall w \in z: x^w \le y^w$ where $z$ is a limit ordinal.

This proves the limit case.