Are there intensional classes independent of the set universe?

The hereditarily finite sets can be regarded as purely extensional sets. Furthermore, they are quite independent of the underlying set universe (at least if we look at them from an extensional point of view). There are probably more purely extensional sets, with similar properties. I just wanted to pin down one concrete example. (From an intensional point of view however, some of these sets might depend strongly on the underlying set universe. Take the natural numbers $\omega$ as an example.)

Do there exists a dual thing for classes, i.e. purely intensional classes? These would be proper classes described intensionally, that are quite independent of the underlying set universe (at least if we look at them from an intensional point of view). A concrete example could be the class of all sets (i.e. the dual notion of the empty set). But what about the class of all sets with two elements? Or the class of all abelian groups? Or the class of all sets different from the sets in a given hereditarily finite set?

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Could you explain a little what purely extensional sets are, and why the hereditarily finite sets are purely extensional, please? – Lawrence Wong Jan 2 '14 at 17:07
@LawrenceWong By purely extensional, I mean that the set can be defined by "listing" its elements, and its elements themselfes are also purely extensional. By "listing" its elements, I mean that the elements will be "the same" (up to isomorphism), independent of the set universe. – Thomas Klimpel Jan 2 '14 at 18:08
I am not sure exactly what you are asking. You seem to give an example of the type of class whose existence you are asking about. – Carl Mummert Jan 3 '14 at 1:11
@CarlMummert The class of all sets certainly is a proper class, but is its "listing" of properties really independent of the set universe? In that case, the "listing" should probably be empty. But are the axioms of ZFC sufficient to ensure this? If not, is it even possible to give a c.e. axiom set that achieves this? If we drop the condition that the class has to be a proper class, we might also ask whether the hereditarily finite sets are purely intensional, i.e. whether the "listing" of their properties is independent of the set universe. I guess they are in ZFC, but that's another question. – Thomas Klimpel Jan 3 '14 at 7:33