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 Th39:
  omega in W implies ex a,M st a is_cofinal_with omega & M = Rank a & M <==> W
proof
  assume omega in W;
  then consider b,M such that
A1: b is_cofinal_with omega & M = Rank b & M is_elementary_subsystem_of
  W by Th35;
  take b,M;
  thus thesis by A1,Th9;
end;
