reserve X,Y,Z for set,
  x,y,z for object,
  E for non empty set,
  A,B,C for Ordinal ,
  L,L1 for Sequence,
  f,f1,f2,h for Function,
  d,d1,d2,d9 for Element of E;

theorem
  (not ex d1 st d1 in d) iff d in Collapse (E,{})
proof
A1: Collapse (E,{}) = { d9 : for d1 st d1 in d9 ex B st B in {} & d1 in
  Collapse (E,B) } by Th1;
  thus (not ex d1 st d1 in d) implies d in Collapse (E,{})
  proof
    assume not ex d1 st d1 in d;
    then for d1 st d1 in d ex B st B in {} & d1 in Collapse (E,B);
    hence thesis by A1;
  end;
  assume d in Collapse (E,{});
  then
A2: ex d9 st d9 = d & for d1 st d1 in d9 ex B st B in {} & d1 in Collapse (E
  ,B) by A1;
  given d1 such that
A3: d1 in d;
  ex B st B in {} & d1 in Collapse (E,B) by A3,A2;
  hence contradiction;
end;
