makarov , I was looking at this and I am not convinced why $\displaystyle \forall z[z \in (\mathcal{F}\cap \mathcal{G})\Rightarrow z=\emptyset]$ is same

as $\displaystyle \mathcal{F} \cap \mathcal{G}\subseteq\{\emptyset\}$.

The first part just means that whenever , $\displaystyle z \in (\mathcal{F}\cap \mathcal{G}) $ , that z has to be an empty set. But the second part means

$\displaystyle \forall z[z \in (\mathcal{F}\cap \mathcal{G})\Rightarrow z \in \{\emptyset\}]$

can you comment ?