reserve A,B for limit_ordinal infinite Ordinal;
reserve B1,B2,B3,B5,B6,D, C for Ordinal;
reserve X for set;
reserve X for Subset of A;
reserve M for non countable Aleph;
reserve X for Subset of M;
reserve N,N1 for cardinal infinite Element of M;

theorem Th22:
  omega in cf M implies for f being sequence of X holds sup rng f in M
proof
  assume
A1: omega in cf M;
  let f be sequence of X;
  per cases;
  suppose
A2: not X = {};
    rng f c= X by RELAT_1:def 19;
    then
A3: rng f c= M by XBOOLE_1:1;
A4: card NAT in cf M by A1;
    dom f = NAT by A2,FUNCT_2:def 1;
    then card rng f c= card NAT by CARD_1:12;
    then card rng f in cf M by A4,ORDINAL1:12;
    hence thesis by A3,CARD_5:26;
  end;
  suppose
    X = {};
    then f = {};
    then sup rng f = {} by ORDINAL2:18,RELAT_1:38;
    hence thesis by ORDINAL1:16,XBOOLE_1:3;
  end;
end;
