reserve V for non empty set,
  A,B,A9,B9 for Element of V;
reserve f,f9 for Element of Funcs(V);
reserve m,m1,m2,m3,m9 for Element of Maps V;
reserve a,b for Object of Ens(V);
reserve f,g,f1,f2 for Morphism of Ens(V);

theorem Th33:
  V <> {{}} & a is terminal implies ex x being set st a = {x}
proof
  assume that
A1: V <> {{}} and
A2: a is terminal;
  set x = the Element of @a;
A3: now
    assume
A4: @a = {};
    now
      consider C being object such that
A5:   C in V and
A6:   C <> {} by A1,ZFMISC_1:35;
      reconsider C as Element of V by A5;
      set b = @C;
      Hom(b,a) <> {} by A2;
      then Funcs(@b,@a) <> {} by Lm6;
      hence contradiction by A4,A6;
    end;
    hence contradiction;
  end;
  now
    assume a <> {x};
    then consider y being object such that
A7: y in @a and
A8: y <> x by A3,ZFMISC_1:35;
    reconsider fy = @a --> y as Function of @a,@a by A7,FUNCOP_1:45;
    reconsider fx = @a --> x as Function of @a,@a by A7,FUNCOP_1:45;
    fx.y = x by A7,FUNCOP_1:7;
    then
A9: fx <> fy by A7,A8,FUNCOP_1:7;
A10: ([[@a,@a],fx]) in Maps(@a,@a) by Th15;
A11: ([[@a,@a],fy]) in Maps(@a,@a) by Th15;
    Maps(@a,@a) c= Maps V by Th17;
    then reconsider
    m1 = [[@a,@a],fx],m2 = [[@a,@a],fy] as Element of Maps(V) by A10,A11;
A12: m2 in Hom(a,a) by A11,Th26;
    m1 in Hom(a,a) by A10,Th26;
    then reconsider f = @m1,g = @m2 as Morphism of a,a by A12,CAT_1:def 5;
    consider h being Morphism of a,a such that
A13: for h9 being Morphism of a,a holds h = h9 by A2;
    f = h by A13
      .= g by A13;
    hence contradiction by A9,XTUPLE_0:1;
  end;
  hence thesis;
end;
