∀X,ℱ(ℱ∩2^X≠∅⇔X∈⟨ℱ⟩X)
$ \forall X,\mathcal F:\mathcal F\cap2^X\neq\varnothing\iff X\in\lang\mathcal F\rang_X
$ \because\forall X,\mathcal F:
$ X\in\lang\mathcal F\rang_X
$ \iff X\in2^X\land\mathcal F\cap2^X\neq\varnothing
$ \underline{\iff\mathcal F\cap2^X\neq\varnothing\quad}_\blacksquare
とくに、$ \mathcal F\subseteq2^Xなら$ \mathcal F\neq\varnothing\iff X\in\lang\mathcal F\rang_X