Chain-meet-closed sets on complete lattices

Let $\mathfrak{A}$ is a complete lattice. I will call a filter base a nonempty subset $T$ of $\mathfrak{A}$ such that $\forall a,b\in T\exists c\in T: (c\le a\wedge c\le b)$. I will call a chain (on $\mathfrak{A}$) a linearly ordered subset of $\mathfrak{A}$.

Now as a part my research of filters I attempt to solve this problem (the problem seems not very difficult and I hope to prove it today or tomorrow, however who knows how difficult it may be):

Definition A subset $S$ of a complete lattice $\mathfrak{A}$ is chain-meet-closed iff for every non-empty chain $T\in\mathscr{P}S$ we have $\bigcap T\in S$.

Conjecture A subset $S$ of a complete lattice $\mathfrak{A}$ is chain-meet-closed iff for every filter base $T\in\mathscr{P}S$ we have $\bigcap T\in S$.