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:

 $\ds \mathbf a + \mathbf 0$ $=$ $\ds \card {A \cup \O}$ Definition of Sum of Cardinals $\ds$ $=$ $\ds \card A + \card \O$ Cardinality of Set Union/Corollary $\ds$ $=$ $\ds \card A$ Cardinality of Empty Set $\ds$ $=$ $\ds \mathbf a$

$\blacksquare$