Entropy bound for a union-closed family

Let $\mathcal{F}$ be a finite union-closed family whose union is nonempty. Some ground element belongs to at least a proportionof its members. Requiring a nonempty union excludes both the empty set family and $\{\varnothing \}$, for which no element can be chosen.

Here is the entropy mechanism. Choose independent uniform random members $A,B$ of $\mathcal{F}$. Then $H(A)=\log |\mathcal{F}|$ and $H(A\cup B)\le \log |\mathcal{F}|$, because the union still belongs to $\mathcal{F}$. If every element had probability at most $u<c$ of belonging to $A$, Will Sawin's binary entropy estimate would givea contradiction when $|\mathcal{F}|>1$. A singleton set family with nonempty union satisfies the conclusion directly.^[ref]