Ordinals have No Zero Divisors

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

Then:


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

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


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

By Ordinal Multiplication by Zero:
 * $\left({ x \cdot 0 }\right) = 0$

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

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

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

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

By Ordinal Multiplication by Zero, both $\left({ 0 \cdot y }\right) = 0$ and $\left({ x \cdot 0 }\right)= 0$.