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 Th25:
  X c= M & card X in cf M implies sup X in M & union X in M
proof
  assume that
A1: X c= M and
A2: card X in cf M;
  set A = order_type_of (RelIncl X);
A3: card A = card X by A1,Th7;
  consider N such that
A4: N c= card A and
A5: A is_cofinal_with N and
  for C st A is_cofinal_with C holds N c= C by Th9;
  sup X is_cofinal_with A by A1,Th6;
  then
A6: sup X is_cofinal_with N by A5,ORDINAL4:37;
A7: now
    assume sup X = M;
    then cf M c= N by A6,Def1;
    hence contradiction by A2,A3,A4,CARD_1:4;
  end;
  for x st x in X holds x is Ordinal by A1;
  then reconsider A = union X as epsilon-transitive epsilon-connected set
by ORDINAL1:23;
A8: sup M = M by ORDINAL2:18;
  sup X c= sup M by A1,ORDINAL2:22;
  then
A9: sup X c< M by A8,A7;
  hence sup X in M by ORDINAL1:11;
  A c= sup X
  proof
    let x be Ordinal;
    assume x in A;
    then consider Y being set such that
A10: x in Y and
A11: Y in X by TARSKI:def 4;
    reconsider Y as Ordinal by A1,A11;
    Y in sup X by A11,ORDINAL2:19;
    then Y c= sup X by ORDINAL1:def 2;
    hence thesis by A10;
  end;
  hence thesis by A9,ORDINAL1:11,12;
end;
