
theorem Lemma16:
  for A being set,
      R being asymmetric Relation of A holds
    R \/ id A is antisymmetric
  proof
    let A be set, R be asymmetric Relation of A;
    for x, y being object st [x,y] in R \/ (id A) &
      [y,x] in R \/ (id A) holds x = y
    proof
      let x,y be object;
      assume
A1:   [x,y] in R \/ (id A) & [y,x] in R \/ (id A); then
Z0:   [x,y] in R or [x,y] in id A by XBOOLE_0:def 3;
Z1:   [y,x] in R or [y,x] in id A by XBOOLE_0:def 3,A1;
      assume x <> y;
      hence thesis by Z1,RELAT_1:def 10,LemAsym,Z0;
    end;
    hence thesis by LemAntisym;
  end;
