
theorem Lemma17:
  for A being set,
      R being asymmetric Relation of A holds
    R misses R~
  proof
    let A be set;
    let R be asymmetric Relation of A;
    for x,y being object holds not [x,y] in R /\ R~
    proof
      let x,y be object;
      assume [x,y] in R /\ R~; then
      [x,y] in R & [x,y] in R~ by XBOOLE_0:def 4; then
      [x,y] in R & [y,x] in R by RELAT_1:def 7;
      hence contradiction by LemAsym;
    end;
    hence thesis by RELAT_1:37,XBOOLE_0:def 7;
  end;
