You're making this harder than it actually is by conflating a bunch of things, comparing apples and oranges. Sets defined by type vs sets defined by lists of properties, specs, rules etc.

For types, just draw the Venn diagrams:

- The set of objects of Type A is entirely within the set of objects of Subtype of A. Subtype of A is the superset.

- the set of things that are Dogs (class or real world) is entirely within the set of things that are Animals.

- the set of things that are Javascript programs is entirely within the set of things that are Typescript programs.