Set is Subset of Intersection of Supersets/Set of Sets
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Theorem
Let $T$ be a set.
Let $\mathbb S$ be a set of sets.
Suppose that for each $S \in \mathbb S$, $T \subseteq S$.
Then:
- $T \subseteq \ds \bigcap \mathbb S$
Proof
Let $x \in T$.
We are given that:
- $\forall S \in \mathbb S: T \subseteq S$
Thus by definition of subset:
- $\forall S \in \mathbb S: x \in S$
Hence by definition of intersection:
- $x \in \ds \bigcap \mathbb S$
Thus by definition of subset:
- $T \subseteq \ds \bigcap \mathbb S$
$\blacksquare$