Exponent Not Equal to Zero

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

Let $x \ne 0$.

Then:


 * $x^y \ne 0$

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

Basis for the Induction

 * $x^0 = 1$ by the definition of ordinal exponentiation.

Therefore, $x^0 \ne 0$.

This proves the basis for the induction.

Induction Step
The inductive hypothesis supposes that $x^y \ne 0$.

This proves the induction step.

Limit Case
The inductive hypothesis says that:


 * $\forall z \in y: x^z \ne 0$

This proves the limit case.