Let $\mathcal{H}$ be a minimal strongly separating system on a base set of size 20. Prove that $\mathcal{H}$ is a Sperner-system.
Let $\mathcal{H}$ be a minimal strongly separating system on a base set of size 25. Show that there exists such that is not a Sperner-system.
Separating system: $\mathcal{T}=\{1,2,...,n\}$, then $\mathcal{H}=\{A|A\in \mathcal{T}\}$ is strongly separating system if $$\forall x,y \in \mathcal{T} \quad \exists A, B \in \mathcal{H}: x\in A, x \notin B \text{ and } y\notin A, y\in B$$
Sperner system: $\forall A, B \in \mathcal{H} \text{ neither } A \subseteq B \text{ nor } B \subseteq A \text{ holds.}$
My question is, that how to check if a separating system is minimal or how to construct one? I think that after that checking if it is a Sperner system should be easy.
Edit 1: $\in$ to $\subseteq$ in Sperner system definition