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

Lemma bigcup_set0_r (T U : Type) (s : set T) (F : T -> set U) :
(forall x, F x <--> set0) ->
\bigcup_(x in s) F x <--> set0.