reserve A, B, X, Y, Z, x, y for set;
reserve f for Function;
reserve O for Ordinal;
reserve a for set;

theorem
  B is infinite implies not B in [:A,B:]
  proof assume that
A1: B is infinite and
A2: B in [:A,B:];
  ex x being object st x in A & B = [x,{x}] by A2,ZFMISC_1:129;
 hence thesis by A1;
end;
