reserve X for set, a,b,c,x,y,z for object;
reserve P,R for Relation;

theorem
  R is_antisymmetric_in X iff R \ id X is_asymmetric_in X
proof
  hereby
    assume
A1: R is_antisymmetric_in X;
    thus R \ id X is_asymmetric_in X
    proof
      let x,y;
      assume that
A2:   x in X and
A3:   y in X and
A4:   [x,y] in R \ id X;
      not [x,y] in id X by A4,XBOOLE_0:def 5;
      then
A5:   x <> y by A2,RELAT_1:def 10;
      [x,y] in R by A4,XBOOLE_0:def 5;
      then not [y,x] in R by A1,A2,A3,A5;
      hence not [y,x] in R \ id X by XBOOLE_0:def 5;
    end;
  end;
  assume
A6: R \ id X is_asymmetric_in X;
  let x,y;
  assume that
A7: x in X & y in X and
A8: [x,y] in R and
A9: [y,x] in R;
  assume
A10: x <> y;
  then not [y,x] in id X by RELAT_1:def 10;
  then
A11: [y,x] in R \ id X by A9,XBOOLE_0:def 5;
  not [x,y] in id X by A10,RELAT_1:def 10;
  then [x,y] in R \ id X by A8,XBOOLE_0:def 5;
  hence contradiction by A6,A7,A11;
end;
