reserve X for set;
reserve UN for Universe;

theorem
  for i being Nat ex x being set st
  x in (sequence_univers).(i+1) \ (sequence_univers).i
  proof
    let i be Nat;
    assume for x be set holds
      not x in (sequence_univers).(i+1) \ (sequence_univers).i;
    then
A1: (sequence_univers).(i+1) \ (sequence_univers).i is empty;
    then
A2: (sequence_univers).(i+1) = (sequence_univers).i by Th105,XBOOLE_1:37;
    per cases;
    suppose
A3:   i = 0;
      GrothendieckUniverse (sequence_univers).0
        = (sequence_univers).(0+1) by Def9
       .= (sequence_univers).0 by A1,A3,Th105,XBOOLE_1:37
       .= {} by Def9;
      hence thesis;
    end;
    suppose i <> 0;
      then consider m be Nat such that
A4:   i = m + 1 by NAT_1:6;
A5:   sequence_univers.(i+1) = UNIVERSE i by Th102;
      sequence_univers.i = UNIVERSE m by A4,Th102;
      then m = i by A2,A5,CLASSES2:71;
      hence thesis by A4;
    end;
  end;
