Since $\linearsystem{A}{\vect{b}}$ has at least one solution, we can apply Theorem PSPHS. Because the solution is assumed to be unique, the null space of $A$ must be trivial. Then Theorem NMTNS implies that $A$ is nonsingular.
The converse of this statement is a trivial application of Theorem NMUS. That said, we could extend our NSMxx series of theorems with an added equivalence for nonsingularity, “Given a single vector of constants, $\vect{b}$, the system $\linearsystem{A}{\vect{b}}$ has a unique solution.”