Intersection is Associative

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Theorem

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

$\displaystyle $	$\displaystyle x \in A \cap \left({B \cap C}\right)$	$\iff$	$\displaystyle x \in A \land \left({x \in B \land x \in C}\right)$	$\displaystyle $	Definition of intersection
$\displaystyle $	$\displaystyle $	$\iff$	$\displaystyle \left({x \in A \land x \in B}\right) \land x \in C$	$\displaystyle $	Rule of Association
$\displaystyle $	$\displaystyle $	$\iff$	$\displaystyle x \in \left({A \cap B}\right) \cap C$	$\displaystyle $	Definition of intersection

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

$\blacksquare$

\(\displaystyle \)	\(\displaystyle x \in A \cap \left({B \cap C}\right)\)	\(\iff\)	\(\displaystyle x \in A \land \left({x \in B \land x \in C}\right)\)	\(\displaystyle \)	Definition of intersection
\(\displaystyle \)	\(\displaystyle \)	\(\iff\)	\(\displaystyle \left({x \in A \land x \in B}\right) \land x \in C\)	\(\displaystyle \)	Rule of Association
\(\displaystyle \)	\(\displaystyle \)	\(\iff\)	\(\displaystyle x \in \left({A \cap B}\right) \cap C\)	\(\displaystyle \)	Definition of intersection