Using Fefferman's analysis of the quantum electron-proton gas, we give a rigorous proof of ionization equilibrium in this system. Ionization equilibrium phases are obtained as low-density and low-temperature limits, letting the chemical potentialmgr(T) approach the ground-state energy of the hydrogen atom as the temperatureT tends to zero. The rate of ionization is determined by the slope ofmgr(T) atT=0 and is correctly given by the Saha formula. We also discuss a simpler model where a single quantum particle interacts with a classical gas of hard spheres.