Intersection is Associative/Family of Sets/Proof 1

Proof
For every $\lambda \in \Lambda$, let $\ds T_\lambda = \bigcap_{i \mathop \in I_\lambda} S_i$.

Then:

Thus:
 * $\ds \bigcap_{i \mathop \in I} S_i = \bigcap_{\lambda \mathop \in \Lambda} T_\lambda = \bigcap_{\lambda \mathop \in \Lambda} \paren {\bigcap_{i \mathop \in I_\lambda} S_i}$