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).

(* Specifications: *)
Lemma subset_eqP A B : reflect (A =i B) (A \subset B \subset A)%set.
Lemma properE A B : A \proper B = (A \subset B) && ~~ (B \subset A).
Lemma eqVproper A B : A \subset B -> A = B \/ A \proper B.
Lemma sub0set A : set0 \subset A.
Lemma imset0 f : imset f set0 = set0.
Lemma imsetS f A A' : A \subset A' -> imset f A \subset imset f A'.