Definition inE := ((in_set0, in_set1, in_setU, in_setI, in_setD), inE).Definition subset_trans B A C := (@le_trans _ _ B A C).Definition proper_trans B A C := (@lt_trans _ _ B A C).Definition sub_proper_trans B A C := (@le_lt_trans _ _ B A C).Definition proper_sub_trans B A C := (@lt_le_trans _ _ B A C).Definition proper_sub A B := (@ltW _ _ A B).

Lemma properE A B : A \proper B = (A \subset B) && ~~ (B \subset A).