theorem
  I is non empty implies not ex X st X is empty-yielding & X is non-empty
proof
  assume
A1: I is non empty;
  given X such that
A2: X is empty-yielding and
A3: X is non-empty;
  consider i being object such that
A4: i in I by A1,XBOOLE_0:def 1;
  X.i is empty by A2,A4;
  hence contradiction by A3,A4;
end;
