  In this note, we investigate partial regularity of weak solutions of the
three dimensional chemotaxis-Navier-Stokes equations, and obtain the
$\frac53$-dimensional Hausdorff measure of the possible singular set is
vanishing at the first blow-up time. The new ingredients are to establish
certain type of local energy inequality and deal with the non-scaling invariant
quantity of $n\ln n$, which seems to be the first description for the singular
set of weak solutions of the chemotaxis-fluid model, which is motivated by
Caffarelli-Kohn-Nirenberg's partial regularity theory \cite{CKN}.
