You can do it in Scala just by forming the intersection type. Subtyping might get in the way (you "lose" types in the intersection which have a subtyping relationship with another) so you can wrap them in an invariant type constructor to work around that.
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The supertype/subset relationship is the most useful thing you get out of this. Most obviously, you can't remove an element from an intersection type, but that's more to do with sets: If you add ten identical Bippys to a set and remove one, is there a Bippy in the set? Who knows?
But I want to experiment with an encoding which is an HList of intersection types, where the set represented by the type is the set of things in the head intersection minus the set of types represented by the tail. The HList gets longer the more times you change the same element.
