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 Th20:
  for M being Aleph for X being Subset of M holds X is unbounded
  implies cf M c= card X
proof
  let M be Aleph;
  let X be Subset of M;
  assume X is unbounded;
  then
A1: sup X = M;
  assume not cf M c= card X;
  then card X in cf M by ORDINAL1:16;
  then sup X in M by CARD_5:26;
  hence contradiction by A1;
end;
