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

theorem Th18:
 {x} << X & y <= x implies {y} << X
proof
  assume A1:{x} << X & y <= x;
  let l,r such that A2: l in {y} & r in X;
  x in {x} by TARSKI:def 1;
  then y < r by A2,A1,Th4;
  hence thesis by A2,TARSKI:def 1;
end;
