This gives you a type like `Inv[A] with Inv[B] with Inv[C]`, and you get a subset test "for free", as it's equivalent to taking the supertype.
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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You probably want to throw another type into the mix as well (IDK, pick one - it doesn't matter as long as you're consistent and it's got no subtype relationship with `Inv`) so that you have a good representation the empty set.
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