Ordinal Addition is Left Cancellable

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

Then:


 * $\paren {z + x} = \paren {z + y} \implies x = y$

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
 * $\paren {1 + \omega} = \paren {0 + \omega}$

but $1 \ne 0$.

Also see

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