reserve X for BCI-algebra;
reserve X1 for non empty Subset of X;
reserve A,I for Ideal of X;
reserve x,y,z for Element of X;
reserve a for Element of A;
reserve X for BCK-algebra;
reserve X for BCI-algebra;
reserve X for BCK-algebra;
reserve I for Ideal of X;

theorem Th35:
  (for I being Ideal of X holds I is commutative Ideal of X) iff {
  0.X} is commutative Ideal of X
proof
  thus (for I being Ideal of X holds I is commutative Ideal of X) implies {0.X
  } is commutative Ideal of X by BCIALG_1:43;
  thus {0.X} is commutative Ideal of X implies for I being Ideal of X holds I
  is commutative Ideal of X
  proof
    assume
A1: {0.X} is commutative Ideal of X;
    let I be Ideal of X;
    for I being Ideal of X holds {0.X} c= I
    proof
      let I be Ideal of X;
      let x be object;
      assume x in {0.X};
      then x=0.X by TARSKI:def 1;
      hence thesis by BCIALG_1:def 18;
    end;
    hence thesis by A1,Th34;
  end;
end;
