reserve U for Universe;
reserve x for Element of U;
reserve U1,U2 for Universe;

theorem
  not [:U,U:] \ [: U\{{}},{{}} :] is Element of U
  proof
    assume
A1: [:U,U:] \ [: U \ { {} }, { {} } :] is Element of U;
    reconsider x = {{}} as Element of U by CLASSES2:56,57;
    now
      let o be object;
      assume o in [:U,{x}:];
      then consider a,b be object such that
A2:   a in U and
A3:   b in {x} and
A4:   o = [a,b] by ZFMISC_1:def 2;
      now
        b = x by A3,TARSKI:def 1;
        hence o in [:U,U:] by A2,A4,ZFMISC_1:def 2;
        now
          assume o in [: U \ { {} } , { {} } :];
          then consider c,d be object such that
          c in U \ {{}} and
A5:       d in {{}} and
A6:       o = [c,d] by ZFMISC_1:def 2;
          c = a & b = d by A4,A6,XTUPLE_0:1;
          hence contradiction by A3,A5,TARSKI:def 1;
        end;
        hence not o in [: U \ { {} } , { {} } :];
      end;
      hence o in [:U,U:] \ [: U \ { {} } , { {} } :] by XBOOLE_0:def 5;
    end;
    then [:U, {x}:] c= [:U,U:] \ [: U \ { {} } , { {} } :] in U by A1;
    then [:U, {x}:] in U by CLASSES4:13;
    then proj1 [:U, {x}:] is Element of U by CLASSES4:36;
    then U in U;
    hence thesis;
  end;
