Sum of Cardinals is Associative

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

Then:
 * $\mathbf a + \left({\mathbf b + \mathbf c}\right) = \left({\mathbf a + \mathbf b}\right) + \mathbf c$

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

Proof
Let $\mathbf a = \operatorname{Card} \left({A}\right)$, $\mathbf b = \operatorname{Card} \left({B}\right)$ and $\mathbf c = \operatorname{Card} \left({C}\right)$ for some sets $A$, $B$ and $C$.

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

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:
 * $\mathbf a + \mathbf b = \operatorname{Card} \left({A \sqcup B}\right) = \operatorname{Card} \left({A \cup B}\right)$
 * $\mathbf b + \mathbf c = \operatorname{Card} \left({B \sqcup C}\right) = \operatorname{Card} \left({B \cup C}\right)$

Then:

Then:

Similarly:

Then:

Finally note that from Union is Associative:
 * $A \cup \left({B \cup C}\right) = \left({A \cup B}\right) \cup C$