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in my previous solution, I made a huge mistake. apologies for that. Here I have proven what you had asked. Doubts will be entertained. Not replying to comment would never help you....

solution.... here note that grad of divergence , divergence of curl and curl of grad is zero ..

see solution attached.

look further

also see it.

solenoidal vectors: The vectors whose curl is zero. here you are asking to find the curl of gradient. but curl of gradient is always zero. In the below proof you can take gradient ...

it's because.....we know divergence simply means flux and when it's 0 means incoming flux = outgoing flux. means nothing inside. and solenoid has nothing inside that's why we calle...

see attached

for solenoid we have to know that divergence of the filed is zero