We consider nonparametric identification of independent private value first-price auction models, in which the analyst only observes winning bids. Our benchmark model assumes an exogenous number of bidders N. We show that, if the bidders observe N, the resulting discontinuities in the winning bid density can be used to identify the distribution of N. The private value distribution can be nonparametrically identified in a second step. This extends, under testable identification conditions, to the case where N is a number of potential buyers, who bid with some unknown probability. Identification also holds in presence of additive unobserved heterogeneity drawn from some parametric distributions. A last class of extensions deals with cartels which can change size across auctions due to varying bidder cartel membership. Identification still holds if the econometrician observes winner identities and winning bids, provided a (unknown) bidder is always a cartel member. The cartel participation probabilities of other bidders can also be identified. An application to USFS timber auction data illustrates the usefulness of discontinuities to analyze bidder participation.