theorem
  not On W is_cofinal_with a
proof
  given psi such that
A1: dom psi = a and
A2: rng psi c= On W and
  psi is increasing and
A3: On W = sup psi;
  On W c= W by ORDINAL2:7;
  then rng psi c= W by A2;
  then sup rng psi in W by A1,Th11,ZF_REFLE:19;
  then sup psi in On W by ORDINAL1:def 9;
  hence contradiction by A3;
end;
