a type-level ordered pair), replacing each element of the field of each in the collection with results in a collection of set pictures isomorphic to the original collection but with their fields disjoint. The union of these set pictures with a new top element yields a set picture whose isomorphism type will have as its preimages under E exactly the elements of the original collection. That is, for any collection of isomorphism types , there is an isomorphism type whose preimage under E is exactly this collection. In particular, there will be an isomorphism type [v] whose preimage under E is the collection of all T[x]'s (including T[v]). Since T[v] E v and E is well-founded, . This resembles the resolution of the Burali–Forti paradox discussed above and in the New Foundations article, and is in fact the local resolution of Mirimanoff's paradox of the set of all well-founded sets. There are ranks of isomorphism classes of set pictures just as there are ranks of sets in the usual set theory. For any collection of set pictures A, define S(A) as the set of all isomorphism classes of set pictures whose preimage under E is a subset of A; call A a "complete" set if every subset of A is a preimage under E. The collection of "ranks" is the smallest collection containing the empty set and closed under the S operation (which is a kind of power set construction) and under unions of its subcollections. It is straightforward to prove (much as in the usual set theory) that the ranks are well-ordered by inclusion, and so the ranks have an index in this well-order: refer to the rank with index as . It is provable that for complete ranks . The union of the complete ranks (which will be the first incomplete rank) with the relation E looks like an initial segment of the universe of Zermelo-style set theory (not necessarily like the full universe of ZFC because it may not be large enough). It is provable that if is the first incomplete rank, then is a complete rank and thus . So there is a "rank of the cumulative hierarchy" with an "external automorphism" T moving the rank downward, exactly the condition on a nonstandard model of a rank in the cumulative hierarchy under which a model of NFU is constructed in the New Foundations article. There are