In this paper, we use the axioms introduced in Eisenberg and Noe (2001) and Rogers and Veraart (2013) and study their consequences in terms of optimal sets of defaulting firms. We show that, from this point of view, the Absolute Priority axiom is not independent. We also show that the optimal sets of defaulting firms characterized in Eisenberg and Noe (2001) are still optimal when the Limited Payment axiom, implicit in Eisenberg and Noe (2001), is further removed. However, some other optimal sets of defaulting firms appear in this case. Finally, with the help of counterexamples, we show that no further weakening in the set of axioms considered can lead to positive results.