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 Th33:
  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
  is_elementary_subsystem_of W
proof
  deffunc D(Ordinal, set) = Rank $1;
  deffunc C(Ordinal,set) = Rank succ $1;
  consider L being Sequence such that
A1: dom L = On W & (0 in On W implies L.0 = Rank 1-element_of W) and
A2: for A st succ A in On W holds L.(succ A) = C(A,L.A) and
A3: for A st A in On W & A <> 0 & A is limit_ordinal holds L.A = D(A,L|
  A) from ORDINAL2:sch 5;
A4: a <> {} & a is limit_ordinal implies L.a = Rank a
  by A3,ZF_REFLE:7;
A5: L.succ a = Rank succ a
  by A2,ZF_REFLE:7;
A6: a <> {} implies L.a = Rank a
  proof
A7: now
A8:   a in On W by ZF_REFLE:7;
      given A such that
A9:   a = succ A;
      A in a by A9,ORDINAL1:6;
      then A in On W by A8,ORDINAL1:10;
      then reconsider A as Ordinal of W by ZF_REFLE:7;
      L.succ A = Rank succ A by A5;
      hence thesis by A9;
    end;
    a is limit_ordinal or ex A st a = succ A by ORDINAL1:29;
    hence thesis by A4,A7;
  end;
  rng L c= W
  proof
    let e be object;
    assume e in rng L;
    then consider u being object such that
A10: u in On W and
A11: e = L.u by A1,FUNCT_1:def 3;
    reconsider u as Ordinal of W by A10,ZF_REFLE:7;
    Rank 1-element_of W in W & Rank u in W by Th31;
    hence thesis by A1,A6,A10,A11;
  end;
  then reconsider L as Sequence of W by RELAT_1:def 19;
  now
    assume {} in rng L;
    then consider e being object such that
A12: e in On W and
A13: {} = L.e by A1,FUNCT_1:def 3;
    reconsider e as Ordinal of W by A12,ZF_REFLE:7;
    e = {} & 1-element_of W = {{}} or e <> {} by ORDINAL3:15;
    then L.e = Rank 1-element_of W & 1-element_of W <> {} or e <> {} & L.e =
    Rank e by A1,A6,ZF_REFLE:7;
    hence contradiction by A13,Th32;
  end;
  then reconsider L as DOMAIN-Sequence of W by A1,RELAT_1:def 9,ZF_REFLE:def 2;
A14: Union L = W
  proof
    thus Union L c= W;
    let e be object;
A15: Union L = union rng L by CARD_3:def 4;
    assume e in W;
    then
A16: e in Rank On W by CLASSES2:50;
    On W is limit_ordinal by CLASSES2:51;
    then consider A such that
A17: A in On W and
A18: e in Rank A by A16,CLASSES1:31;
    reconsider A as Ordinal of W by A17,ZF_REFLE:7;
    L.A = Rank A & L.A in rng L by A1,A6,A17,A18,CLASSES1:29,FUNCT_1:def 3;
    then Rank A c= Union L by A15,ZFMISC_1:74;
    hence thesis by A18;
  end;
A19: 0-element_of W in On W by ZF_REFLE:7;
A20: for a,b st a in b holds L.a c= L.b
  proof
    let a,b;
    assume
A21: a in b;
    then
A22: Rank a in Rank b & succ a c= b by CLASSES1:36,ORDINAL1:21;
A23: L.b = Rank b by A6,A21;
    L.a = Rank a or a = 0-element_of W & L.a = Rank 1-element_of W &
    1-element_of W = succ 0-element_of W by A1,A19,A6;
    hence thesis by A22,A23,CLASSES1:37,ORDINAL1:def 2;
  end;
A24: for a st a <> {} & a is limit_ordinal holds L.a = Union (L|a)
  proof
    let a;
    assume that
A25: a <> {} and
A26: a is limit_ordinal;
A27: a in On W by ZF_REFLE:7;
A28: L.a = Rank a by A4,A25,A26;
    thus L.a c= Union (L|a)
    proof
      let e be object;
      assume e in L.a;
      then consider B such that
A29:  B in a and
A30:  e in Rank B by A25,A26,A28,CLASSES1:31;
      B in On W by A27,A29,ORDINAL1:10;
      then reconsider B as Ordinal of W by ZF_REFLE:7;
A31:  succ B in On W & Union (L|a) = union rng (L|a) by CARD_3:def 4,ZF_REFLE:7
;
      L.succ B = Rank succ B by A5;
      then
A32:  Rank B c= L.succ B by CLASSES1:33;
      succ B in a by A26,A29,ORDINAL1:28;
      then L.succ B c= Union (L|a) by A1,A31,FUNCT_1:50,ZFMISC_1:74;
      then Rank B c= Union (L|a) by A32;
      hence thesis by A30;
    end;
    let e be object;
    assume e in Union (L|a);
    then e in union rng (L|a) by CARD_3:def 4;
    then consider X such that
A33: e in X and
A34: X in rng (L|a) by TARSKI:def 4;
    consider u being object such that
A35: u in dom (L|a) and
A36: X = (L|a).u by A34,FUNCT_1:def 3;
    reconsider u as Ordinal by A35;
A37: X = L.u by A35,A36,FUNCT_1:47;
A38: dom (L|a) c= a by RELAT_1:58;
    then u in On W by A27,A35,ORDINAL1:10;
    then reconsider u as Ordinal of W by ZF_REFLE:7;
    L.u c= L.a by A20,A35,A38;
    hence thesis by A33,A37;
  end;
  assume omega in W;
  then consider phi such that
A39: phi is increasing & phi is continuous and
A40: for a st phi.a = a & {} <> a holds L.a is_elementary_subsystem_of
  Union L by A20,A24,Th30;
  take phi;
  thus phi is increasing & phi is continuous by A39;
  let a,M;
  assume that
A41: phi.a = a and
A42: {} <> a and
A43: M = Rank a;
  M = L.a by A6,A42,A43;
  hence thesis by A40,A14,A41,A42;
end;
