Sum of Cardinals is Commutative

Theorem
Let $\mathbf a$ and $\mathbf b$ be cardinals.

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

where $\mathbf a + \mathbf b$ denotes the sum of $\mathbf a$ and $\mathbf b$.

Proof
Let $\mathbf a = \operatorname{Card} \left({A}\right)$ and $\mathbf b = \operatorname{Card} \left({B}\right)$ for some sets $A$ and $B$ such that $A \cap B = \varnothing$.

Then: