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;

theorem Th24:
  for A holds A is limit_ordinal iff for C st C in A holds succ C in A
proof
  let A;
  thus A is limit_ordinal implies for C st C in A holds succ C in A
  proof
    assume A is limit_ordinal;
    then
A1: A = union A;
    let C;
    assume C in A;
    then consider z such that
A2: C in z and
A3: z in A by A1,TARSKI:def 4;
    for b be object holds b in { C } implies b in z by A2,TARSKI:def 1;
    then
A4: { C } c= z;
A5: z is Ordinal by A3,Th9;
    then C c= z by A2,Def2;
    then succ C c= z by A4,XBOOLE_1:8;
    then succ C = z or succ C c< z;
    then
A6: succ C = z or succ C in z by A5,Th7;
    z c= A by A3,Def2;
    hence thesis by A3,A6;
  end;
  assume
A7: for C st C in A holds succ C in A;
  now
    let a be object;
     reconsider aa=a as set by TARSKI:1;
    assume
A8: a in A;
    then a is Ordinal by Th9;
    then
A9: succ aa in A by A7,A8;
    a in succ aa by Th2;
    hence a in union A by A9,TARSKI:def 4;
  end;
  then
A10: A c= union A;
  now
    let a be object;
    assume a in union A;
    then consider z such that
A11: a in z and
A12: z in A by TARSKI:def 4;
    z c= A by A12,Def2;
    hence a in A by A11;
  end;
  then union A c= A;
  then A = union A by A10;
  hence thesis;
end;
