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 & X in W implies ex M st X in M & M in W & M is
  being_a_model_of_ZF
proof
  assume
A1: omega in W;
A2: W = Rank On W by CLASSES2:50;
  assume X in W;
  then the_rank_of X in the_rank_of W by CLASSES1:68;
  then the_rank_of X in On W by A2,CLASSES1:64;
  then reconsider a = the_rank_of X as Ordinal of W by ZF_REFLE:7;
  consider b,M such that
A3: a in b and
A4: M = Rank b and
A5: M <==> W by A1,Th38;
  take M;
  X c= Rank a by CLASSES1:def 9;
  then
A6: X in Rank succ a by CLASSES1:32;
  succ a c= b by A3,ORDINAL1:21;
  then Rank succ a c= M by A4,CLASSES1:37;
  hence X in M by A6;
  b in On W by ZF_REFLE:7;
  hence M in W by A2,A4,CLASSES1:36;
  W is being_a_model_of_ZF by A1,ZF_REFLE:6;
  then W |= ZF-axioms by Th4;
  then M |= ZF-axioms by A5,Th8;
  hence thesis by A4,Th5;
end;
