Answer by Pierre Castéran for How to to use the fact that combining these hypothesis is false in Coq

If you choose to use gt_not_le_stt, you don't need to destruct a nor y.

  Goal forall a y, a > y -> a <= y -> False.Proof. intros a y H H0. apply Arith_prebase.gt_not_le_stt in H.  exact (H H0).Qed.

Of course, there are a lot of other solutions, depending on the libraries, tactics and lemmas you choose to use (including the magic lia).