reserve k,n,m for Nat,
  A,B,C for Ordinal,
  X for set,
  x,y,z for object;
reserve f,g,h,fx for Function,
  K,M,N for Cardinal,
  phi,psi for
  Ordinal-Sequence;
reserve a,b for Aleph;

theorem Th21:
  cf omega = omega
proof
  defpred P[set,set] means $2 c= $1;
  assume
A1: cf omega <> omega;
  cf omega c= omega by Def1;
  then cf omega in omega by A1,CARD_1:3;
  then reconsider B = cf omega as finite set;
  set n = card B;
A2: for x,y being set st P[x,y] & P[y,x] holds x = y;
A3: for x,y,z being set st P[x,y] & P[y,z] holds P[x,z] by XBOOLE_1:1;
  omega is_cofinal_with n by Def1;
  then consider xi being Ordinal-Sequence such that
A4: dom xi = n and
A5: rng xi c= omega and
  xi is increasing and
A6: omega = sup xi;
  reconsider rxi = rng xi as finite set by A4,FINSET_1:8;
A7: rxi <> {} by A6,ORDINAL2:18;
  consider x being set such that
A8: x in rxi & for y being set st y in rxi & y <> x holds not P[y,x]
    from CARD_2:sch 3(A7,A2,A3);
  reconsider x as Ordinal by A5,A8;
  now
    let A;
    assume A in rng xi;
    then A c= x or not x c= A by A8;
    hence A in succ x by ORDINAL1:22;
  end;
  then
A9: omega c= succ x by A6,ORDINAL2:20;
  succ x in omega by A5,A8,ORDINAL1:28;
  hence contradiction by A9;
end;
