Demonstration that electric current at equilibrium is zero in crystals

As it is well known, electrons at equilibrium (no external field) do not conduct electric current, i.e.

$int_{BZ} dk,v_{k},f(epsilon_k)=0$

where $f(epsilon_k)$ is the Fermi-Dirac distribution

$f(epsilon)=frac{1}{e^{beta(epsilon-mu)}+1}$

$v_k$ is the velocity

$v_k=frac{partial epsilon_k}{partial k}$

and $BZ$ is the Brillouin zone.

Is it possible to demonstrate that just using the explicit expression
of the Fermi-Dirac distribution and the fact that we are considering functions periodic on the reciprocal lattice? I mean, is it possible to say that the integral is zero just from the general properties of the quantities involved? (for example, in the free-electron case it comes from the fact that it is the integral of an odd function in $k$).