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 Th24:
  omega in cf M & X is unbounded implies for B1 st B1 in M ex B st
  B in M & B1 in B & B in limpoints X
proof
  defpred P[set,set,set] means $2 in $3;
  assume
A1: omega in cf M;
  assume
A2: X is unbounded;
  then reconsider X1= X as non empty Subset of M by Th7;
  let B1 such that
A3: B1 in M;
  reconsider LB1 = LBound(B1,X1) as Element of X1;
A4: for n being Nat for x being Element of X1 ex y being Element
  of X1 st P[n,x,y]
  proof
    let n be Nat;
    let x be Element of X1;
    reconsider x1=x as Element of M;
    succ x1 in M by ORDINAL1:28;
    then consider y being Ordinal such that
A5: y in X1 and
A6: succ x1 c= y by A2,Th6;
    reconsider y1=y as Element of X1 by A5;
    take y1;
    x in succ x1 by ORDINAL1:6;
    hence thesis by A6;
  end;
  consider L being sequence of X1 such that
A7: L.0 = LB1 and
A8: for n be Nat holds P[n,L.n,L.(n+1)] from RECDEF_1:sch 2(
  A4);
  set L2=L;
  reconsider LB2=LB1 as Element of M;
A9: dom L = NAT by FUNCT_2:def 1;
  then
A10: L.0 in rng L by FUNCT_1:def 3;
  then
A11: LB2 in sup rng L by A7,ORDINAL2:19;
A12: for C st C in rng L2 ex D st D in rng L2 & C in D
  proof
    let C;
    assume C in rng L2;
    then consider C1 being object such that
A13: C1 in dom L2 and
A14: C = L2.C1 by FUNCT_1:def 3;
    reconsider C2=C1 as Element of NAT by A13,FUNCT_2:def 1;
    L2.(C2+1) in X;
    then reconsider C3=L2.(C2+1) as Element of M;
    take C3;
    thus C3 in rng L2 by A9,FUNCT_1:def 3;
    thus thesis by A8,A14;
  end;
A15: rng L c= X by RELAT_1:def 19;
  then rng L c= M by XBOOLE_1:1;
  then reconsider
  SUP = sup rng L as limit_ordinal infinite Element of M by A1,A10,A12,Th3,Th22
;
  take SUP;
A16: sup ( X /\ SUP) = SUP
  proof
    assume sup ( X /\ SUP) <> SUP;
    then consider B5 such that
A17: B5 in SUP and
A18: (X /\ SUP) c= B5 by Th5;
    consider B6 being Ordinal such that
A19: B6 in rng L and
A20: B5 c= B6 by A17,ORDINAL2:21;
    B6 in sup rng L by A19,ORDINAL2:19;
    then B6 in (X /\ SUP) by A15,A19,XBOOLE_0:def 4;
    then B6 in B5 by A18;
    then B6 in B6 by A20;
    hence contradiction;
  end;
  B1 in LB2 by A2,A3,Th9;
  hence thesis by A16,A11,ORDINAL1:10;
end;
