Intersection of Subsets is Subset/Set of Sets
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Theorem
Let $T$ be a set.
Let $\mathbb S$ be a non-empty set of sets.
Suppose that for each $S \in \mathbb S$:
- $S \subseteq T$
Then:
- $\bigcap \mathbb S \subseteq T$
Proof
Let $x \in \bigcap \mathbb S$.
Then by the definition of intersection:
- $\forall S \in \mathbb S: x \in S$.
Since $\mathbb S$ is non-empty by the premise, it has some element $S$.
Then $x \in S$.
Since $S \in \mathbb S$, the premise shows that $S \subseteq T$.
By the definition of subset, $x \in T$.
Since this holds for each $x \in \bigcap \mathbb S$:
- $\bigcap \mathbb S \subseteq T$
$\blacksquare$