Proper names are usually supposed to be rigid de jure. Given this claim and some other usual assumptions—namely that properties are explicated as intensions of a sort and that various possible worlds have various universes—one may derive the following inconsistent pair of conclusions: (i) for all properties P and for all possible worlds w it holds that an object, o, exemplifies P with respect to w only if o exists in w; and (ii) there is at least one property P and at least one possible world w such that o exemplifies P with respect to w even though o fails to exist in w. The aim of the present paper is to show how the problematic pair of conclusions is derived, spell out its background (most notably the idea of rigidity de jure) and review possible ways out.