We investigate a fully discrete finite element scheme for the three dimensional incompressible magnetohydrodynamic problem based on magnetic vector potential formulation. The formulation enjoys the novel feature that it can always produce an exactly divergence-free magnetic induction discretized solution. Using a mixed finite element approach, we discretize the model by the fully discrete semi-implicit Euler scheme with the velocity and the pressure approximated by stable MINI finite elements and the magnetic vector potential by Nedelec edge elements. Convergence analysis and error estimates for the velocity and the magnetic vector potential are rigorously established. Finally, several numerical experiments are presented to illustrate the convergence properties of the numerical scheme.