Union is Associative/Family of Sets

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

Let $\displaystyle I = \bigcup_{\lambda \mathop \in \Lambda} I_\lambda$ denote the union of $\family {I_\lambda}_{\lambda \mathop \in \Lambda}$.

Then:
 * $\displaystyle \bigcup_{i \mathop \in I} S_i = \bigcup_{\lambda \mathop \in \Lambda} \paren {\bigcup_{i \mathop \in I_\lambda} S_i}$

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

Then:

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

Also see

 * General Associativity of Set Intersection