reserve A,B,C,O for Ordinal,
        X for set,
        o for object,
        x,y,z,t,r,l for Surreal;

theorem Th21:
  {x} << {y} iff x < y
proof
  hereby assume A1: {x} << {y};
    x in {x} & y in {y} by TARSKI:def 1;
    hence x<y by A1;
  end;
  assume A2:x < y;
  let l,r;
  assume l in {x} & r in {y};
  then l=x & r=y by TARSKI:def 1;
  hence thesis by A2;
end;
