# Set is Subset of Intersection of Supersets/General Result

## Theorem

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

Let $X$ be a set such that:

$\forall i \in I: X \subseteq S_i$

Then:

$X \subseteq \bigcup_{i \mathop \in I} S_i$

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

## Proof

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

Then:

 $\displaystyle x$ $\in$ $\displaystyle X$ $\displaystyle \leadsto \ \$ $\, \displaystyle \forall i \in I: \,$ $\displaystyle x$ $\in$ $\displaystyle S_i$ Definition of Subset $\displaystyle \leadsto \ \$ $\, \displaystyle \forall i \in I: \,$ $\displaystyle x$ $\in$ $\displaystyle \bigcap_{i \mathop \in I} S_i$ Definition of Intersection of Family $\displaystyle \leadsto \ \$ $\displaystyle X$ $\subseteq$ $\displaystyle \bigcap_{i \mathop \in I} S_i$ Definition of Subset

$\blacksquare$