Note as to inclusion-minimal non-Bondy systems

Let $S$ be a finite set, $s=\mathrm{\mid }S\mathrm{\mid }\ge 6$. Given a non-negative integer $t$, there exists an inclusion-minimal non-Bondy system $\mathcal{A}$ of size $t$ on $S$ if and only if $s+1\le t\le 2s$.