This letter deals with the design and analysis of a stabilizing model predictive control (MPC) scheme for discrete-time switched linear systems with at least one stabilizable subsystem. An approach for pruning the switching sequences is proposed which is used for reducing the online computation and ensuring stability of the MPC scheme. With the pruning approach, the value function of the MPC scheme becomes an exponential Lyapunov function (ELF). Consequently, the MPC scheme results in globally exponentially stable (GES) closed-loop dynamics for switched linear systems for all prediction horizons. Feasibility of the MPC scheme is discussed and simulation results are given for illustrating the effectiveness of the MPC scheme.