Ordinal Addition is Left Cancellable

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

Then:


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

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.

Note that:

However:

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

Warning
Ordinal addition is not right-cancellable.

$\left({ 1 + \omega }\right) = \left({ 0 + \omega }\right)$ by Natural Number Plus Ordinal, but $1 \ne 0$.

Also see

 * Natural Number Addition is Cancellable, a result showing that natural number addition is both left and right cancellable.