(** [list_disjoint l1 l2] holds iff [l1] and [l2] have no elements
  in common. *)

Definition list_disjoint (A: Type) (l1 l2: list A) : Prop :=
  forall (x y: A), In x l1 -> In y l2 -> x <> y.

Lemma list_disjoint_notin: forall (A: Type) (l1 l2: list A) (a: A), list_disjoint l1 l2 -> In a l1 -> ~(In a l2).