The reaction is:
HA <-> H+ + A-
pKa is -log[H+] for the concentration of the acid when the acid/base equilibrium is established. if pH > pKa, the concentration of [H+] is lower than the equilibrium concentration and the reaction will move to the right.
pKb applies to the conjugate base - if pOH > pKb the base will be protonating since [OH] is too low.
Added: If you want a more precies proof, you should consider that Ka=[H][A]/[HA] at equilibrium. But for [HA]>>[H] we have that [A]/[HA] = 1, and Ka = [H] at equilibrium or pKa = pH at equilibrium.
The interesting assumption here is that [HA] >> [H], which is correct most of the time. It is still means that at the extreme cases the above will be incorrect.