Ordering on 1-Based Natural Numbers is Total Ordering

Theorem
Let $\N_{> 0}$ be the $1$-based natural numbers.

Let $<$ be the strict ordering on $\N_{>0}$.

Then $<$ is a (strict) total ordering.

Proof
From Ordering on $1$-Based Natural Numbers is Trichotomy we have that $<$ is trichotomy.

From Ordering on $1$-Based Natural Numbers is Transitive we have that $<$ is transitive.

From Trichotomy is Antireflexive it follows that $<$ is antireflexive.

It follows by definition that $<$ is a strict ordering.

By the trichotomy law it follows that $<$ is a strict total ordering.