reserve X,Y,Z,X1,X2,X3,X4,X5,X6 for set, x,y for object;
reserve a,b,c for object, X,Y,Z,x,y,z for set;
reserve A,B,C,D for Ordinal;
reserve F,G for Function;
reserve L,L1 for Sequence;

theorem Th32:
  for D ex A st D in A & A is limit_ordinal
proof
  let D;
  consider Field being set such that
A1: D in Field and
A2: for X,Y holds X in Field & Y c= X implies Y in Field and
A3: for X holds X in Field implies bool X in Field and
  for X holds X c= Field implies X,Field are_equipotent or X in Field
  by ZFMISC_1:112;
  for X st X in On Field holds X is Ordinal & X c= On Field
  proof
    let X;
    assume
A4: X in On Field;
    then reconsider A = X as Ordinal by Def9;
A5: A in Field by A4,Def9;
    thus X is Ordinal by A4,Def9;
    let y be object;
    assume
A6: y in X;
    then y in A;
    then reconsider B = y as Ordinal by Th9;
    B c= A by A6,Def2;
    then B in Field by A2,A5;
    hence thesis by Def9;
  end;
  then reconsider ON = On Field as epsilon-transitive epsilon-connected set
         by Th15;
  take ON;
  thus D in ON by A1,Def9;
  A in ON implies succ A in ON
  proof
A7: succ A c= bool A
    proof
      let x be object;
       reconsider xx=x as set by TARSKI:1;
      assume x in succ A;
      then x in A or x = A by Th4;
      then xx c= A by Def2;
      hence thesis;
    end;
    assume A in ON;
    then A in Field by Def9;
    then bool A in Field by A3;
    then succ A in Field by A2,A7;
    hence thesis by Def9;
  end;
  hence thesis by Th24;
end;
