Union is Associative/Family of Sets

Theorem
Let $\left \langle{S_i}\right \rangle_{i \in I}$ and $\left \langle{I_\lambda}\right \rangle_{\lambda \in \Lambda}$ be indexed families of sets.

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

Then:
 * $\displaystyle \bigcup_{i \mathop \in I} S_i = \bigcup_{\lambda \mathop \in \Lambda} \left({\bigcup_{i \mathop \in I_\lambda}}\right) S_i$

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

Then:

Thus:
 * $\displaystyle \bigcup_{i \mathop \in I} S_i = \bigcup_{\lambda \mathop \in \Lambda} T_\lambda = \bigcup_{\lambda \mathop \in \Lambda} \left({\bigcup_{i \mathop \in I_\lambda}}\right) S_i$