
theorem
  for X being set, R being Relation of X
  holds R is irreflexive iff id X misses R
proof
  let X be set, R be Relation of X;
  hereby
    assume A1: R is irreflexive;
    assume id X meets R;
    then (id X) /\ R <> {} by XBOOLE_0:def 7;
    then consider z being object such that
      A2: z in (id X) /\ R by XBOOLE_0:def 1;
    consider x,y being object such that
      A3: z = [x,y] by A2, RELAT_1:def 1;
    A4: [x,y] in id X & [x,y] in R by A2, A3, XBOOLE_0:def 4;
    then x = y & x in field R by RELAT_1:def 10, RELAT_1:15;
    hence contradiction by A1, A4, RELAT_2:def 2, RELAT_2:def 10;
  end;
  assume A5: id X misses R;
  field R c= X \/ X
    by RELSET_1:8;
  then id field R misses R by A5, FUNCT_4:3, XBOOLE_1:63;
  hence thesis by RELAT_2:2;
end;
