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 $\mathbf b$.

Let $\mathbf a = \map \Card A$ and $\mathbf b = \map \Card B$ for some sets $A$ and $B$ such that $A \cap B = \O$.

Then:

$\ds \mathbf a + \mathbf b$

$=$

$\ds \map \Card {A \cup B}$

$\ds $

$=$

$\ds \map \Card {B \cup A}$

$\ds $

$=$

$\ds \mathbf b + \mathbf a$

$\blacksquare$