(* Picks A_i for i\in I *)
 
(** - so, given an index set $I$ and an indexing function $f$ mapping 
 this set to the part_set (powerset) of X,  (ie. $f(i) = A_i\ (\in (part\_set A))$,) 
 then (indexed_intersection f) will be $\bigcap_{i\in I} A_i$. *)

Definition indexed_intersection : part_set A.

Lemma not_in_part_then_not_in_indexed_intersection : forall a : A, (exists i : I, ~ in_part a (f i)) -> ~ in_part a indexed_intersection.