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;

theorem Th6:
  X is unbounded iff for B1 st B1 in A ex C st C in X & B1 c= C
proof
  thus X is unbounded implies for B1 st B1 in A ex C st C in X & B1 c= C
  proof
    assume
A1: X is unbounded;
    let B1;
    assume B1 in A;
    then not X c= B1 by A1,Th4;
    then consider x being object such that
A2: x in X and
A3: not x in B1;
    reconsider x1 = x as Element of A by A2;
    take x1;
    thus x1 in X by A2;
    thus thesis by A3,ORDINAL1:16;
  end;
  assume
A4: for B1 st B1 in A ex C st C in X & B1 c= C;
  assume X is bounded;
  then consider B1 such that
A5: B1 in A and
A6: X c= B1 by Th4;
  consider C such that
A7: C in X and
A8: B1 c= C by A4,A5;
  X c= C by A6,A8;
  then C in C by A7;
  hence contradiction;
end;
