Identity of Cardinal Sum is Zero

Theorem
Let $\mathbf a$ be a cardinal.

Then:
 * $\mathbf a + \mathbf 0 = \mathbf a$

where $\mathbf a + \mathbf 0$ denotes the sum of the zero cardinal and $\mathbf a$.

That is, $\mathbf 0$ is the identity element of the sum operation on cardinals.

Proof
Let $\mathbf a = \card A$ for some set $A$.

From Union with Empty Set we have $A \cup \O = A$.

From Intersection with Empty Set we have $A \cap \O = \O$.

So $A$ and $\O$ are disjoint and so: