Union is Associative

Theorem
Let $$A, B$$ and $$C$$ be sets.

Then:
 * $$A \cup \left({B \cup C}\right) = \left({A \cup B}\right) \cup C$$

where $$\cup$$ denotes set union.

Proof
$$ $$ $$

Therefore, $$x \in A \cup \left({B \cup C}\right)$$ iff $$x \in \left({A \cup B}\right) \cup C$$.

Thus it has been shown that $$A \cup \left({B \cup C}\right) = \left({A \cup B}\right) \cup C$$.