Is cross product of del, $\nabla \times \nabla$, zero in vectors?

Question

I came across a vector term like

$$ \nabla \cdot (\nabla \times \mathbf{F}) = 0 $$

So I though to solve it like

\begin{align*} M &= \nabla \cdot (\nabla \times \mathbf{F}) \\ &= \mathbf{F} \cdot (\nabla \times \nabla) - \nabla \cdot (\nabla \times \mathbf{F}) \\ &= \mathbf{F} \cdot (\nabla \times \nabla) - M, \\ \vphantom{\Big(} 2M &= \mathbf{F} \cdot (\nabla \times \nabla) = 0. \end{align*}

Here I suppose $\nabla \times \nabla$ must be zero. So

$$ \nabla \cdot (\nabla \times \mathbf{F}) = 0 $$

Is this is true that $\nabla \times \nabla$ (This is meaningless)?

Answer 1

From Wikipedia,

If two vectors have the same direction (or have the exact opposite direction from one another, i.e. are not linearly independent) or if either one has zero length, then their cross product is zero.

Answer 2

Is this is true that ∇ x ∇ = 0 ?

Yes!

Read more in Wikipedia.