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

theorem
  [{},Day A] in Day succ A\Day A & [Day A,{}] in Day succ A\Day A
proof
  A1:{} << Day A <<{};
  o in {}\/Day A implies ex O st O in succ A & o in Day O
    by ORDINAL1:6;
  then A2:[{},Day A] in Day (succ A) & [Day A,{}] in Day (succ A)
    by A1,SURREAL0:46;
  then reconsider eD=[{},Day A],De =[Day A,{}] as Surreal;
  A3:eD <= eD & De <=De;
  A4: eD in {eD} & De in {De} by TARSKI:def 1;
  L_De << {De} & {eD} << R_eD by Th11;
  then not eD in Day A & not De in Day A by A3,A4;
  hence thesis by A2,XBOOLE_0:def 5;
end;
