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
  not A is limit_ordinal iff ex B st A = succ B
proof
  thus not A is limit_ordinal implies ex B st A = succ B
  proof
    assume not A is limit_ordinal;
    then consider B such that
A1: B in A and
A2: not succ B in A by Th24;
    take B;
    assume
A3: A <> succ B;
    succ B c= A by A1,Th17;
    then succ B c< A by A3;
    hence contradiction by A2,Th7;
  end;
  given B such that
A4: A = succ B;
  B in A & not succ B in A by A4,Th2;
  hence thesis by Th24;
end;
