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

Lemma setU_l_subset {U} (s1 s2 s3 : set U) :
s1 \subset s3 ->
s2 \subset s3 ->
(s1 :|: s2) \subset s3.