Intersection is Associative

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

Proof: $$x\in A \cap (B\cap C)\Longleftrightarrow (x\in A)\wedge (x\in B\cap C)$$ by Definition of $$\cap$$ $$\Longleftrightarrow (x\in A) \wedge ((x\in B)\wedge (x\in C))$$ by Definition of $$\cap$$ $$\Longleftrightarrow ((x\in A) \wedge (x\in B))\wedge x\in C)$$ by Associativity of $$\wedge$$ $$\Longleftrightarrow (x\in A\cap B)\wedge (x\in C)$$ by Definition of $$\cap$$ $$\Longleftrightarrow x\in (A\cap B)\cap C$$ by Definition of $$\cap$$ Therefore, $$x \in A \cap (B\cap C)$$ if and only if $$x \in (A \cap B) \cap C$$

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

Q.E.D.