Uniqueness of equilibria in first-price auction with discrete valuations
Ruling out (strictly) overlapping intervals should be possible using the fact that players must be indifferent among all actions in the support of the mixed strategy, or that these type of auctions is efficient. 

Uniqueness is tricky, simply because there are many possible interval structures like that one.  But I guess you mean, equilibria are always of this form. In that case, uniqueness should be possible among symmetric equilibria, after ruling out the overlapping intervals, you need to show that the support of the mixed strategy is convex (which should be the case, otherwise money is left on the table), that the intervals are ordered (which comes from efficiency),  and put restrictions on the bidders with the lowest valuation. 

Without symmetry, you can have situations where players with the same valuation use different mixed strategies  (even with different support), so the problem there seems untractable to me.