Definition has_unique_least_element (A:Type) (R:A->A->Prop) (P:A->Prop) :=
  exists! x, P x /\ forall x', P x' -> R x x'.

Lemma dec_inh_nat_subset_has_unique_least_element : forall P:nat->Prop, (forall n, P n \/ ~ P n) -> (exists n, P n) -> has_unique_least_element le P.