 Remark 2.5  Note that if \(v\not\in\Pi_i(w)\) then \(w\not\in \Pi_i(v)\). Hence, by property 1, \(w\not\preceq_i v\) and \(v\not\preceq_i w\). Thus, we have the following equivalence: \(v\in \Pi_i(w)\) iff \(w\preceq_i v\) or \(v\preceq_i w\).