reserve H,S for ZF-formula,
  x for Variable,
  X,Y for set,
  i for Element of NAT,
  e,u for set;
reserve M,M1,M2 for non empty set,
  f for Function,
  v1 for Function of VAR,M1,
  v2 for Function of VAR,M2,
  F,F1,F2 for Subset of WFF,
  W for Universe,
  a,b,c for Ordinal of W,
  A,B,C for Ordinal,
  L for DOMAIN-Sequence of W,
  va for Function of VAR,L.a,
  phi,xi for Ordinal-Sequence of W;
reserve psi for Ordinal-Sequence;

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;
