This paper aims to provide arguments for the claim that ellipsis licensing requires elided material to constitute a subset of its antecedent. On the empirical side, I focus on deriving Ross’ (1970) generalization that backward gapping is restricted to OV contexts. A new operation Total Impoverishment is proposed for ellipsis, which involves insertion of null morphemes into the ellipsis site in a late insertion framework such as Distributed Morphology. This approach is developed as a possible alternative to Merchant’s (2001) account based on the formal [E]-feature. It will be shown that the correlation between word order and the directionality of gapping falls out under the present approach, and in addition more general arguments for the role of the subset in ellipsis will be presented, as well as a critique of the [E]-feature approach from the perspective of alternatives to formal features.