reserve A,A1,A2,B,C,D for Ordinal,
  X,Y for set,
  x,y,a,b,c for object,
  L,L1,L2,L3 for Sequence,
  f for Function;

theorem
  A in X implies inf X in X
proof
  defpred P[Ordinal] means $1 in X;
  assume A in X;
  then
A1: ex A st P[A];
  consider A such that
A2: P[A] & for B st P[B] holds A c= B from ORDINAL1:sch 1(A1);
A3: A in On X by A2,ORDINAL1:def 9;
A4: now
    let x be object;
    thus x in A implies for Y st Y in On X holds x in Y
    proof
      assume
A5:   x in A;
      let Y;
      assume
A6:   Y in On X;
      then reconsider B = Y as Ordinal by ORDINAL1:def 9;
      Y in X by A6,ORDINAL1:def 9;
      then A c= B by A2;
      hence thesis by A5;
    end;
    assume for Y st Y in On X holds x in Y;
    hence x in A by A3;
  end;
  On X <> 0 by A2,ORDINAL1:def 9;
  hence thesis by A2,A4,SETFAM_1:def 1;
end;
