Fermi energy for two spin states equal in equilibrium?

[lampCable]

Homework Statement

Let's consider conduction electrons (at T=0K) that are put in a magnetic field. The electrons can have spin that is parallel or antiparallel to the magnetic field. Below is the density of occupied states for such a system (horizontally) as a function of energy (vertically), similar to a figure shown in my textbook.

My textbook claims that, in thermal equilibrium, the Fermi energy for both spin states must be equal, but I do not fully understand why this is the case. I think it has to do with the system being in its lowest energy state, and I reason as follows:

The Fermi energy is the energy required to add another electron to the given spin state. So if the Fermi energies are not equal then some electrons in one of the spin states would have energy greater than the Fermi energy of the other spin state. These electrons would spontaneously change spin state because it would lower the energy in the system, thus increasing the total entropy. As more electrons change spin state the Fermi energies are "evened out", so at equilibrium the Fermi energies must be equal.

Is this a correct interpretation as to why the Fermi energies are equal?

Homework Equations

The Attempt at a Solution

 

[TSny]

I think your argument is good.