Let $B$ be the reduced row-echelon form of the augmented matrix $\augmented{A}{\vect{b}}$. Because the system $\linearsystem{A}{\vect{b}}$ is consistent, we know by Theorem RCLS that the last column of $B$ is not a pivot column. Suppose now that $r<n$. Then by Theorem FVCS the system would have infinitely many solutions. From this contradiction, we see that $r=n$ and the first $n$ columns of $B$ are each pivot columns. Then the sequence of row operations that converts $\augmented{A}{\vect{b}}$ to $B$ will also convert $A$ to $I_n$. Applying Theorem NMRRI we conclude that $A$ is nonsingular.