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

theorem
  R is irreflexive iff id field R misses R
proof
  hereby
    assume R is irreflexive;
    then
A1: R is_irreflexive_in field R;
    now
      let a,b be object;
      assume
A2:   [a,b] in id (field R) /\ R;
      then [a,b] in id (field R) by XBOOLE_0:def 4;
      then
A3:   a in field R & a = b by RELAT_1:def 10;
      [a,b] in R by A2,XBOOLE_0:def 4;
      hence contradiction by A1,A3;
    end;
    hence id (field R) misses R by RELAT_1:37,XBOOLE_0:def 7;
  end;
  assume
A4: id (field R) misses R;
  let a;
  assume a in field R;
  then [a,a] in id field R by RELAT_1:def 10;
  hence thesis by A4,XBOOLE_0:3;
end;
