
theorem Lemma21:
  for X being set, R being asymmetric Relation of X holds
    R misses id X
  proof
    let X be set, R be asymmetric Relation of X;
    for x,y being object holds not [x,y] in R /\ id X
    proof
      let x,y be object;
A0:   x in field R implies not [x,x] in R by RELAT_2:def 2,def 10;
      assume [x,y] in R /\ id X; then
      [x,y] in R & [x,y] in id X by XBOOLE_0:def 4;
      hence contradiction by A0,RELAT_1:def 10,RELAT_1:15;
    end;
    hence thesis by XBOOLE_0:def 7,RELAT_1:37;
  end;
