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 Th28:
  M is Mahlo implies M is limit_cardinal
proof
  assume M is Mahlo;
  then
A1: { N : N is regular } is_stationary_in M;
  then reconsider REG={N : N is regular} as Subset of M;
  assume not M is limit_cardinal;
  then consider M1 being Cardinal such that
A2: nextcard M1 = M by CARD_1:def 4;
  M1 in M by A2,CARD_1:18;
  then succ M1 in M by ORDINAL1:28;
  then M \ succ M1 is closed unbounded by Th12;
  then REG /\ (M \ succ M1) <> {} by A1;
  then consider M2 being object such that
A3: M2 in REG /\ (M \ succ M1) by XBOOLE_0:def 1;
  M2 in REG by A3,XBOOLE_0:def 4;
  then consider N such that
A4: N = M2 and
  N is regular;
  M2 in (M \ succ M1) by A3,XBOOLE_0:def 4;
  then not N in succ M1 by A4,XBOOLE_0:def 5;
  then not N c= M1 by ORDINAL1:22;
  then M1 in N by ORDINAL1:16;
  then N in M & nextcard M1 c= N by CARD_3:90;
  then N in N by A2;
  hence contradiction;
end;
