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
  On X <> 0 & (for A st A in X holds D c= A) implies D c= inf X
proof
  assume that
A1: On X <> 0 and
A2: for A st A in X holds D c= A;
  let x be object such that
A3: x in D;
  for Y st Y in On X holds x in Y
  proof
    let Y;
    assume
A4: Y in On X;
    then reconsider A = Y as Ordinal by ORDINAL1:def 9;
    A in X by A4,ORDINAL1:def 9;
    then D c= A by A2;
    hence thesis by A3;
  end;
  hence thesis by A1,SETFAM_1:def 1;
end;
