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

theorem
  R is_asymmetric_in X implies R \/ id X is_antisymmetric_in X
proof
  assume
A1: R is_asymmetric_in X;
  let x,y;
  assume that
A2: x in X & y in X and
A3: [x,y] in R \/ id X and
A4: [y,x] in R \/ id X;
  assume
A5: x <> y;
  then not [y,x] in id X by RELAT_1:def 10;
  then
A6: [y,x] in R by A4,XBOOLE_0:def 3;
  not [x,y] in id X by A5,RELAT_1:def 10;
  then [x,y] in R by A3,XBOOLE_0:def 3;
  hence contradiction by A1,A2,A6;
