Intersection is Associative

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

Then $$A \cap \left({B \cap C}\right) = \left({A \cap B}\right) \cap C$$.

Proof
Therefore, $$x \in A \cap \left({B \cap C}\right)$$ if and only if $$\left({x \in A \cap B}\right) \cap C$$.

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

Q.E.D.