Qausi-static displacement current (Purcell)

 P: 9 Hello all, On page 329, chapter 9 of Purcell's E&M book, he describes why the "displacement current" produces nearly zero magnetic field for slowly varying fields. By taking the curl of the displacement current Jd, he shows that curlJd=-(1/4$\pi$c)(∂2B/∂t2), so that for slowly varying fields, curlJd is nearly zero. From this, using a symmetry/superposition argument he concludes that there can be no magnetic field for such a current distribution. It is this last step that I do not understand; how does curlJd=0 imply B=0? Thanks!