Vanishing set and order reversing

Let \(A\) be a commutative ring, then the Vanishing set reverses subsets, i.e. for \(T \subseteq S \subseteq A\) it holds

\begin{align*} V(T) \supseteq V(S) \end{align*}

Let \(\mathfrak{p} \in V(S)\), then by definition \(S \subseteq \mathfrak{p}\) is true and therefore also

\begin{align*} T \subseteq \mathfrak{p} \end{align*}

Hence \(\mathfrak{p} \in V(T)\)