Corollary 2

A set $\Phi$ of filters on a non-empty set $X$ has a least upper bound in the set of all filters on $X$ if and only if, for all finite sequences $(\mathfrak{F}_i)_{1 \leq i \leq n}$ of elements of $\Phi$ and all $A_i \in \mathfrak{F}_i$ ($1 \leq i \leq n$), the intersection $A_1 \cap \cdots \cap A_n$ is not empty.

For this condition expresses that the union $\mathfrak{G}$ of the filters $\mathfrak{F} \in \Phi$ satisfies the condition of Proposition 1.