Intersection is Associative

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

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

where $$\cap$$ denotes set intersection.

Proof
$$ $$ $$

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

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