Intersection Operation on Supersets of Subset is Closed

Theorem
Let $S$ be a set.

Let $T \subseteq S$ be a given subset of $S$.

Let $\powerset S$ denote the power set of $S$

Let $\mathscr S$ be the subset of $\powerset S$ defined as:
 * $\mathscr S = \set {Y \in \powerset S: T \subseteq Y}$

Then the algebraic structure $\struct {\mathscr S, \cap}$ is closed.

Proof
Let $A, B \in \mathscr S$.

We have that:

and:

Thus we have:

Hence the result by definition of closed algebraic structure.

Also see

 * Union Operation on Supersets of Subset is Closed