Intersection is Associative

Theorem

$A \cap \paren {B \cap C} = \paren {A \cap B} \cap C$

Family of Sets

Let $\family {S_i}_{i \mathop \in I}$ and $\family {I_\lambda}_{\lambda \mathop \in \Lambda}$ be indexed families of sets.

Let $\displaystyle I = \bigcap_{\lambda \mathop \in \Lambda} I_\lambda$.

Then:

$\displaystyle \bigcap_{i \mathop \in I} S_i = \bigcap_{\lambda \mathop \in \Lambda} \paren {\bigcap_{i \mathop \in I_\lambda} S_i}$

Proof

 $\displaystyle$  $\displaystyle x \in A \cap \paren {B \cap C}$ $\displaystyle$ $\leadstoandfrom$ $\displaystyle x \in A \land \paren {x \in B \land x \in C}$ Definition of Set Intersection $\displaystyle$ $\leadstoandfrom$ $\displaystyle \paren {x \in A \land x \in B} \land x \in C$ Rule of Association: Conjunction $\displaystyle$ $\leadstoandfrom$ $\displaystyle x \in \paren {A \cap B} \cap C$ Definition of Set Intersection

Therefore:

$x \in A \cap \paren {B \cap C}$ if and only if $x \in \paren {A \cap B} \cap C$

Thus it has been shown that:

$A \cap \paren {B \cap C}\ = \paren {A \cap B} \cap C$

$\blacksquare$