Cardinal Zero is Less than Cardinal One

Theorem
The zero cardinal $0$ is less than one:
 * $0 < 1$

Proof
We have that the Cardinals are Totally Ordered.

Let $\mathcal R \subseteq \O \times \set \O$ be any arbitrary relation between $\O$ and $\set \O$.

We have that $\mathcal R$ is vacuously many-to-one.

Also vacuously, $\mathcal R$ is left-total.

Thus by definition, $\mathcal R$ is in fact a mapping.

From Empty Mapping is Unique, this relation $\mathcal R$ is the unique mapping from $\O$ to $\set \O$.

Also vacuously, $\mathcal R$ is an injection.

So, by definition, $\set \O$ dominates $\O$.

Since Empty Set is Subset of All Sets, we have $\O \subseteq \set \O$.

Hence the result.