Producing Gauss's law for B field in differential form

Gauss's law for B field in integral form:

(1)
Applying Gauss's Thm:

(2)

(3)
Assume that:

(4)
suggesting the existance of a magnetic monopole. Then a volume could be choosen to enclosed only one such monopole. In that case:

(5)
However, this consequence contradicts the previous assertion. So,

(4)
must be false. Therefore,

(6)