P1) a∈X olsun.
a∈X⇒X⊈
P_2) A\in\mathcal{P} ve B\subseteq A olsun.
\left.\begin{array}{rr} A\in \mathcal{P}\Rightarrow A\subseteq X\setminus\{a\} \\ \\ B\subseteq A \end{array}\right\}\Rightarrow B\subseteq X\setminus\{a\}\Rightarrow B\in\mathcal{P}.
P_3) A\notin\mathcal{P} ve B\notin\mathcal{P} olsun.
\left.\begin{array}{rr} A\notin\mathcal{P}\Rightarrow A\nsubseteq X\setminus \{a\}\Rightarrow a\in A \\ \\ B\notin\mathcal{P}\Rightarrow B\nsubseteq X\setminus \{a\}\Rightarrow a\in B \end{array}\right\}\Rightarrow A\cap B\nsubseteq X\setminus \{a\}\Rightarrow A\cap B\notin \mathcal{P}.