Intersection is Largest Subset/Family of Sets

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Theorem

Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$.


Then for all sets $X$:

$\displaystyle \paren {\forall i \in I: X \subseteq S_i} \iff X \subseteq \bigcap_{i \mathop \in I} S_i$

where $\displaystyle \bigcap_{i \mathop \in I} S_i$ is the intersection of $\family {S_i}$.


Proof

Let $X \subseteq S_i$ for all $i \in I$.

Then from Set is Subset of Intersection of Supersets: General Result:

$\displaystyle X \subseteq \bigcap_{i \mathop \in I} S_i$

$\Box$


Now suppose that $\displaystyle X \subseteq \bigcap_{i \mathop \in I} S_i$.

From Intersection is Subset: Family of Sets we have:

$\forall i \in I: \bigcap_{j \mathop \in I} S_j \subseteq S_i$

So from Subset Relation is Transitive, it follows that:

$\displaystyle \forall i \in I: X \subseteq \bigcap_{j \mathop \in I} S_j \subseteq S_i$

So it follows that $\forall i \in I: X \subseteq S_i$.

So:

$\displaystyle X \subseteq \bigcap_{i \mathop \in I} S_i \implies \paren {\forall i \in I: X \subseteq S_i}$

$\Box$


Hence:

$\displaystyle \paren {\forall i \in I: X \subseteq S_i} \iff X \subseteq \bigcap_{i \mathop \in I} S_i$

$\blacksquare$


Also see


Sources