Ordinals have No Zero Divisors

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


 * $\displaystyle \left({ x \cdot y }\right) = 0 \iff \left({ x = 0 \lor y = 0 }\right)$

Necessary Condition
Suppose that $\left({ x \cdot y }\right) = 0$ and that $x \ne 0$.


 * $\displaystyle x \ne 0 \implies 0 < x$

But also $\left({ x \cdot 0 }\right) = 0$ by Ordinal Multiplication by Zero.

Therefore $\left({ x \cdot y }\right) = \left({ x \cdot 0 }\right)$

By Ordinal Multiplication is Left Cancellable, we have that $y = 0$.

Sufficient Condition
If $x = 0$, then $\left({ x \cdot y }\right) = \left({ 0 \cdot y }\right)$ by Leibniz's Rule.

If $y = 0$, then $\left({ x \cdot y }\right) = \left({ x \cdot 0 }\right)$ by Leibniz's Rule.

Both $\left({ 0 \cdot y }\right)$ and $\left({ x \cdot 0 }\right)$ equal $0$ by Ordinal Multiplication by Zero.