Strict Ordering on Integers is Trichotomy

Theorem
Let $\eqclass {a, b} {}$ and $\eqclass {c, d} {}$ be integers, as defined by the formal definition of integers.

Then exactly one of the following holds:

That is, strict ordering is a trichotomy.

Proof
By the formal definition of integers, we have that $a, b, c, d, e, f$ are all natural numbers.

To eliminate confusion between integer ordering and the ordering on the natural numbers, let $a \preccurlyeq b$ denote that the natural number $a$ is less than or equal to the natural number $b$.

We have:

Then:

Similarly:

Then:

and:

This demonstrates that $<$, $=$ and $>$ are mutually exclusive.

Now:

Similarly:

and:

demonstrating that either $<$, $=$ or $>$ must hold.