Set is Subset of Intersection of Supersets/General Result

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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

$\blacksquare$


Sources