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Set Theory Regulatory Axiom and Ranks

  1. May 1, 2008 #1
    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
  2. jcsd
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