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

theorem
  R is antisymmetric iff R /\ (R~) c= id (dom R)
proof
A1: now
    assume R is antisymmetric;
    then
A2: R is_antisymmetric_in field R;
    now
      let a,b be object;
      assume
A3:   [a,b] in R /\ (R~);
      then [a,b] in R~ by XBOOLE_0:def 4;
      then
A4:   [b,a] in R by RELAT_1:def 7;
      then
A5:   b in dom R by XTUPLE_0:def 12;
A6:   [a,b] in R by A3,XBOOLE_0:def 4;
      then a in field R & b in field R by RELAT_1:15;
      then a = b by A2,A6,A4;
      hence [a,b] in id (dom R) by A5,RELAT_1:def 10;
    end;
    hence R /\ (R~) c= id (dom R);
  end;
  now
    assume
A7: R /\ (R~) c= id (dom R);
    now
      let a,b;
      assume that
      a in field R and
      b in field R and
A8:   [a,b] in R and
A9:   [b,a] in R;
      [a,b] in R~ by A9,RELAT_1:def 7;
      then [a,b] in R /\ (R~) by A8,XBOOLE_0:def 4;
      hence a = b by A7,RELAT_1:def 10;
    end;
    hence R is antisymmetric by Def4;
  end;
  hence thesis by A1;
end;
