Sum of Cardinals is Associative

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

Then:
 * $\mathbf a + \paren {\mathbf b + \mathbf c} = \paren {\mathbf a + \mathbf b} + \mathbf c$

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

Proof
Let $\mathbf a = \card A, \mathbf b = \card B$ and $\mathbf c = \card C$ for some sets $A$, $B$ and $C$.

Let $A, B, C$ be pairwise disjoint, that is:
 * $A \cap B = \O$
 * $B \cap C = \O$
 * $A \cap C = \O$

Then we can define:


 * $A \sqcup B := A \cup B$
 * $B \sqcup C := B \cup C$
 * $A \sqcup C := A \cup C$

where $A \sqcup B$ denotes the disjoint union of $A$ and $B$.

Then we have:
 * $\mathbf a + \mathbf b = \card {A \sqcup B} = \card {A \cup B}$
 * $\mathbf b + \mathbf c = \card {B \sqcup C} = \card {B \cup C}$

Then:

Then:

Similarly:

Then:

Finally note that from Union is Associative:
 * $A \cup \paren {B \cup C} = \paren {A \cup B} \cup C$