Ordinal Multiplication is Left Cancellable

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

Let $z \ne 0$.

Then:


 * $\displaystyle \left({ z \cdot x }\right) = \left({ z \cdot y }\right) \implies x = y$

Proof
For the proof, $<$, $\in$, and $\subsetneq$ will be used interchangeably. This is justified by Transitive Set is Proper Subset of Ordinal iff Element of Ordinal.

Since $z \ne 0$ and $z \not < 0$, $0 < z$ by Ordinal Membership Trichotomy.

Note that:

In addition:

This contradicts the consequents of the first two equations, so:

Also see

 * Natural Number Multiplication is Cancellable