reserve v,x,x1,x2,x3,x4,y,y1,y2,y3,y4,z,z1,z2 for object,
  X,X1,X2,X3,X4,Y,Y1,Y2,Y3,Y4,Y5,
  Z,Z1,Z2,Z3,Z4,Z5 for set;
reserve p for pair object;

theorem
  X <> {} implies ex v st v in X & not ex x,y st (x in X or y in X) & v
  = [x,y]
proof
  assume X <> {};
  then consider Y such that
A1: Y in X and
A2: not ex Y1 st Y1 in Y & not Y1 misses X by XREGULAR:2;
  take v = Y;
  thus v in X by A1;
  given x,y such that
A3: x in X or y in X and
A4: v = [x,y];
A5: { x,y } in Y by A4,TARSKI:def 2;
  x in { x,y } & y in { x,y } by TARSKI:def 2;
  hence contradiction by A2,A5,A3,XBOOLE_0:3;
end;
