Instance proper_set_incl A :
Morphisms.Proper (set_eq ==> set_eq ==> Basics.impl) (@set_incl A).
Instance eq_setI U : Proper (set_eq ==> set_eq ==> set_eq) (@setI U).

Lemma setI_subset_compat {U} (s1 s2 s1' s2' : set U) :
s1 \subset s1' ->
s2 \subset s2' ->
(s1 :&: s2) \subset (s1' :&: s2').