Union is Associative

Theorem
Let $$A, B$$ and $$C$$ be sets. Then $$A \cup (B\cup C) = (A\cup B)\cup C$$.

Direct 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 $$A \cup \left({B \cup C}\right) = \left({A \cup B}\right) \cup C$$.