Ordinal Addition is Left Cancellable

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

Then:


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

That is, ordinal addition is left cancellable.

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.

By Finite Ordinal Plus Transfinite Ordinal
 * $\left({1 + \omega}\right) = \left({0 + \omega}\right)$

but $1 \ne 0$.

Also see

 * Natural Number Addition is Cancellable: natural number addition is both left and right cancellable.