Why does the chlor alkali process work?

Chlorine is a stronger oxidiser than oxygen, and sodium is a much stronger reducing agent than hydrogen. Therefore, in the chlor alkali process, when voltage is applied to the electrodes and the solution is one of brine, why don't we end up electrolysing the water instead? We should, in fact, start seeing chlorine gas formation only when the brine is extremely thick, almost like a semi solid.


