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_set0 {U} (s1 s2 : set U) :
(forall x, s1 x -> ~ s2 x) ->
(s1 :&: s2) <--> set0.