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