reserve N for Cardinal;
reserve M for Aleph;
reserve X for non empty set;
reserve Y,Z,Z1,Z2,Y1,Y2,Y3,Y4 for Subset of X;
reserve S for Subset-Family of X;
reserve x for set;
reserve F,Uf for Filter of X;
reserve S for non empty Subset-Family of X;
reserve I for Ideal of X;
reserve S,S1 for Subset-Family of X;
reserve FS for non empty Subset of Filters(X);
reserve X for infinite set;
reserve Y,Y1,Y2,Z for Subset of X;
reserve F,Uf for Filter of X;
reserve x for Element of X;

theorem Th27:
  M is strong_limit implies M is limit_cardinal
proof
  assume
A1: M is strong_limit;
  assume not M is limit_cardinal;
  then consider N such that
A2: M = nextcard N;
  M c= exp(2,N) by A2,Th24;
  then
A3: not exp(2,N) in M by CARD_1:4;
  N in M by A2,CARD_1:18;
  hence contradiction by A1,A3;
end;
