# Set Theory Regulatory Axiom and Ranks

1. May 1, 2008

### moo5003

1. The problem statement, all variables and given/known data

Assume that D is a transitive set. Let B be a set with the property that for any a in D, a is a subset of B implies a is an element of B.

Show that D is a subset of B.

3. The attempt at a solution

My first step is to show that the empty set must be an element of D if D is non-empty. (D = 0 is an easy case since its trivially a subset of B)

By the regularity axiom there must be some element c such that

c intersect D = empty

But since D is transitive it must contain all elements of c thus c must be empty.

Since the empty set is a subset of B it implies the empty set is and element of B since the empty set is an element of D.

I'm a little stuck on how to continue from here, any ideas?

Last edited: May 1, 2008