reserve A,X,X1,X2,Y,Y1,Y2 for set, a,b,c,d,x,y,z for object;
reserve P,P1,P2,Q,R,S for Relation;

theorem
  R is empty-yielding iff for X st X in rng R holds X = {}
proof
  thus R is empty-yielding implies
  for X st X in rng R holds X = {} by TARSKI:def 1;
  assume
A1: for X st X in rng R holds X = {};
  let Y be object;
  assume Y in rng R;
  then Y = {} by A1;
  hence thesis by TARSKI:def 1;
end;
