Hint Resolve @join_joins @join_joins' @join_join_sub @join_join_sub'.
Definition nonidentity {A} `{Perm_alg A} (a: A) := ~(identity a).

Lemma join_sub_antisym {A} {JA: Join A}{PA: Perm_alg A}{SA: Sep_alg A}: forall x y,
join_sub x y ->
join_sub y x ->
x = y.