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 Th3:
  (X is non empty & for B1 st B1 in X ex B2 st B2 in X & B1 in B2 )
  implies sup X is limit_ordinal infinite Ordinal
proof
  assume X is non empty;
  then
A1: ex x being object st x in X by XBOOLE_0:def 1;
  assume
A2: for B1 st B1 in X ex B2 st B2 in X & B1 in B2;
A3: for C st C in sup X holds succ C in sup X
  proof
    let C;
    assume C in sup X;
    then consider B3 such that
A4: B3 in X and
A5: C c= B3 by ORDINAL2:21;
    consider B2 such that
A6: B2 in X and
A7: B3 in B2 by A2,A4;
    C in B2 by A5,A7,ORDINAL1:12;
    then
A8: succ C c= B2 by ORDINAL1:21;
    B2 in sup X by A6,ORDINAL2:19;
    hence thesis by A8,ORDINAL1:12;
  end;
  X c= sup X by Th2;
  then reconsider SUP = sup X as non empty limit_ordinal Ordinal by A3,A1,
ORDINAL1:28;
  SUP is limit_ordinal infinite Ordinal;
  hence thesis;
end;
