Set Intersection Preserves Subsets/Families of Sets/Corollary

Theorem
Let $I$ be an indexing set.

Let $\left \langle {B_\alpha} \right \rangle_{\alpha \mathop \in I}$ be an indexed family of subsets of a set $S$.

Let $A$ be a set such that $A \subseteq B_\alpha$ for all $\alpha \in I$.

Then:


 * $\displaystyle A \subseteq \bigcap_{\alpha \mathop \in I} B_\alpha$

Proof
For each $\alpha \in I$, define $A_\alpha := A$.

Then by Intersection is Idempotent, it follows that:


 * $\displaystyle \bigcap_{\alpha \mathop \in I} A_\alpha = A$

Since $A \subseteq B_\alpha$ for all $\alpha \in I$, the premises of Set Intersection Preserves Subsets are satisfied.

Applying this theorem gives:


 * $\displaystyle A = \bigcap_{\alpha \mathop \in I} A_\alpha \subseteq \bigcap_{\alpha \mathop \in I} B_\alpha$

which is precisely the desired result.