Proof of the Fourier Transform of 1 (in the distribution sense)

Question

I am aware of the standard way to prove this but was wondering if this would work as well:

$\rm TF[1]\,=\,TF[1]\ast \delta\,=\,2\pi TF \left[1\times TF^{-1}[\delta]\right]\,=\,2\pi\delta$