Instance eq_bigcap T U : Proper (set_eq ==> pointwise_relation T (@set_eq U) ==> set_eq) bigcap.

Lemma setI_set_incl :
forall (A : Type) (s1 s2 s3 : set A),
s1 \subset s2 ->
s1 \subset s3 ->
s1 \subset s2 :&: s3.