# Intersection is Associative/Family of Sets/Proof 2

## Theorem

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

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

Then:

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

## Proof

 $\ds \bigcap_{i \mathop \in I} S_i$ $=$ $\ds \map \complement {\map \complement {\bigcap_{i \mathop \in I} S_i} }$ Complement of Complement $\ds$ $=$ $\ds \map \complement {\bigcup_{i \mathop \in I} \map \complement {S_i} }$ De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Intersection $\ds$ $=$ $\ds \map \complement {\bigcup_{\lambda \mathop \in \Lambda} \paren {\bigcup_{i \mathop \in I_\lambda} \map \complement {S_i} } }$ General Associativity of Set Union $\ds$ $=$ $\ds \map \complement {\bigcup_{\lambda \mathop \in \Lambda} \paren {\map \complement {\bigcap_{i \mathop \in I_\lambda} S_i} } }$ De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Intersection $\ds$ $=$ $\ds \bigcap_{\lambda \mathop \in \Lambda} \paren {\map \complement {\map \complement {\bigcap_{i \mathop \in I_\lambda} S_i} } }$ De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Union $\ds$ $=$ $\ds \bigcap_{\lambda \mathop \in \Lambda} \paren {\bigcap_{i \mathop \in I_\lambda} S_i}$ Complement of Complement

$\blacksquare$