Intersection is Associative

Theorem
Set intersection is associative:


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

Proof
$$ $$ $$

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

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

Also see

 * Union is Associative
 * Set Difference is Not Associative
 * Symmetric Difference is Associative