In this paper, we consider a fractional Nernst-Planck-Poisson-Navier-Stokes system in ${\mathbb{R}}^3$. First, we obtain a priori estimates by using energy estimates. Then, we construct an iterative solution sequence by solving the approximate problem and obtaining the local existence and uniqueness of the classical solution. Finally, combining the local existence with a priori estimates, the global existence and uniqueness of the classical solution with small initial data are obtained.