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:

$T \subseteq \ds \bigcap \mathbb S$

$\blacksquare$