reserve X,Y,Z for set,
        x,y,z for object,
        A,B,C for Ordinal;
reserve U for Grothendieck;

theorem Th17: :: Theorem 5
  for U be Grothendieck holds U is Tarski
proof
  let U be Grothendieck;
  thus U is subset-closed;
  thus for X be set holds X in U implies bool X in U by Def1;
  let X be set such that
A1: X c= U;
  not X in U implies X,U are_equipotent
  proof
    assume not X in U;
    then consider f be Function such that
A2: f is one-to-one & dom f = On U & rng f = X by A1,Th16;
    not U in U;
    then consider g be Function such that
A3: g is one-to-one & dom g = On U & rng g = U by Th16;
    set gf = g*(f");
    dom (f") = X & rng (f") = On U by A2,FUNCT_1:33;
    then dom gf = X & rng gf = U by A3,RELAT_1:27,28;
    hence thesis by A2,A3,WELLORD2:def 4;
  end;
  hence thesis;
end;
