Ordinal Addition is Left Cancellable

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


 * $\displaystyle ( z + x ) = ( z + y ) \implies x = y$

Proof
For the proof, $<$, $\in$, and $\subsetneq$ will be used interchangeably. This is justified by Ordinal Proper Subset Membership.