Fredholm alternative — the two faces of the Fredholm determinant

For a homogeneous Fredholm integral equation with separable kernel $x(t)=\lambda\int_a^bk(t,s)x(s)\,ds$, consider the following statements: (I) If Fredholm determinant $D(\lambda)\neq0$, then integral equation has only trivial solution i.e. $x(t)=0$. (II) If Fredholm determinant $D(\lambda)=0$, then integral equation has infinitely many solutions.