  The crystalline Chern classes of the value of a locally free crystal vanish
on a smooth variety defined over a perfect field. Out of this we conclude new
cases of de Jong's conjecture relating the geometric \'etale fundamental group
of a smooth projective variety defined over a perfect field and the triviality
of its category of isocrystals. We also discuss the case of the Gau{\ss}-Manin
convergent isocrystal.
