induction ls; intuition; simpl in *.
destruct (eqd a2 a); subst.
intuition.
simpl in *.
intuition.
eapply IHls; eauto.
Qed.

Lemma removeFirst_NoDup_not_in : forall (A : Set)(eqd : eq_dec A)(ls : list A)(a : A),
NoDup ls ->
~In a (removeFirst eqd ls a).