Union is Associative

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

Proof: $$x\in A \cup (B\cup C)\Longleftrightarrow (x\in A)\vee (x\in B\cup C)$$ by Definition of $$\cup$$ $$\Longleftrightarrow (x\in A) \vee ((x\in B)\vee (x\in C))$$ by Definition of $$\cup$$ $$\Longleftrightarrow ((x\in A) \vee (x\in B))\vee (x\in C)$$ by Associativity of $$\vee$$ $$\Longleftrightarrow ((x\in A) \vee (x\in B))\vee (x\in C)$$ by Definition of $$\cup$$ $$\Longleftrightarrow (x\in A\cup B)\vee (x\in C)$$ by Definition of $$\cup$$ $$\Longleftrightarrow x\in (A\cup B)\cup C$$ by Definition of $$\cup$$

Thus, it has been shown that $$A \cup (B\cup C) = (A\cup B)\cup C$$

QED