reserve H,S for ZF-formula,
  x for Variable,
  X,Y for set,
  i for Element of NAT,
  e,u for set;
reserve M,M1,M2 for non empty set,
  f for Function,
  v1 for Function of VAR,M1,
  v2 for Function of VAR,M2,
  F,F1,F2 for Subset of WFF,
  W for Universe,
  a,b,c for Ordinal of W,
  A,B,C for Ordinal,
  L for DOMAIN-Sequence of W,
  va for Function of VAR,L.a,
  phi,xi for Ordinal-Sequence of W;
reserve psi for Ordinal-Sequence;

theorem
  omega in W implies ex phi st phi is increasing & phi is continuous &
  for a,M st phi.a = a & {} <> a & M = Rank a holds M <==> W
proof
  assume omega in W;
  then consider phi such that
A1: phi is increasing & phi is continuous and
A2: for a,M st phi.a = a & {} <> a & M = Rank a holds M
  is_elementary_subsystem_of W by Th33;
  take phi;
  thus phi is increasing & phi is continuous by A1;
  let a,M;
  assume phi.a = a & {} <> a & M = Rank a;
  hence thesis by A2,Th9;
end;
