code poetry


RANKX-Funktion (DAX): RANKX(<table>, <expression>[, <value>[, <order>[, <ties>]]])
Proving rank(℘(x))=rank(x)+
Enderton defines the rank of a set A to be the least ordinal α such that A⊆Vα (equivalently, A∈Vα+). He the derives the following identity: rank(A)=⋃{(rank(x))+:x∈A} for all sets A.
In Exercise 30 of Chapter 7, the reader is asked to prove several identities involving rank. For instance, that rank{a,b}=max(rank(a),rank(b))+. I am having trouble proving the second identity, that rank(℘(x))=rank(x)+ for all sets x. I am not sure whether I am missing some elementary identity that would let me prove the identity, or whether I am misunderstanding the definition of rank.
Clearly, z∈rank(℘(x))⇔(∃y)(y⊆x∧(z∈rank(y)∨z=rank(y)). On the other hand z∈(rank(x))+⇔(∃y)(y∈x∧z∈(rank(y))+)∨z=rank(x). I'm not sure why the first statement should imply the second, and conversely, however.
>> elementary-set-theory

Now isn’t that poetic…