We present a cross-correlation of the European Large Area Infrared Space Observatory (ISO) survey (ELAIS) with the ELAIS Deep X-ray Survey of the N1 and N2 fields. There are seven Chandra point sources with matches in the ELAIS Final Analysis 15{mu}m catalogue, out of a total of 28 extragalactic ISO sources present in the Chandra fields. Five of these are consistent with active galactic nuclei (AGN) giving an AGN fraction of ~19 per cent in the 15{mu}m flux range 0.8-6mJy. We have co-added the hard X-ray fluxes of the individually undetected ISO sources and find a low significance detection consistent with star formation in the remaining population. We combine our point source cross-correlation fraction with the XMM-Newton observations of the Lockman Hole and Chandra observations of the Hubble Deep Field North to constrain source count models of the mid-infrared galaxy population. The low dust-enshrouded AGN fraction in ELAIS implied by the number of cross-identifications between the ELAIS mid-infrared sample and the Chandra point sources is encouraging for the use of mid-infrared surveys to constrain the cosmic star formation history, provided there are not further large undetected populations of Compton-thick AGN.