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_impl_l {T} (s1 s2 : set T) : s1 \subset s2 -> s1 :&: s2 <--> s1.