Proof. Let $Q \in \text{Proj}(A)$ with $0 \le Q \le P$, then by the comparability theorem, either $Q \preceq P - Q$ or $P - Q \preceq Q$. Assume without loss of generality that $Q \preceq P - Q$.

Let $V \in A$ such that $Q = V^{*}V$ and $VV^{*} \le P - Q$, then $V$ is a partial isometry with initial and final spaces contained in $P(H)$. Since $P$ is abelian, $V \in PAP$, and $Q = V^{*}V = VV^{*} \le P - Q$. Therefore $Q = 0$, and $P$ is minimal.$\square$