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 Th34:
  omega in W implies ex b,M st a in b & M = Rank b & M
  is_elementary_subsystem_of W
proof
  assume
A1: omega in W;
  then consider phi such that
A2: phi is increasing & phi is continuous and
A3: for a,M st phi.a = a & {} <> a & M = Rank a holds M
  is_elementary_subsystem_of W by Th33;
  consider b such that
A4: a in b and
A5: phi.b = b by A1,A2,Th28;
  reconsider M = Rank b as non empty set by A4,CLASSES1:36;
  take b,M;
  thus thesis by A3,A4,A5;
end;
