Annie's wrong to be so silly (5)

I believe the answer is:

inane

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'so silly' is the definition.
'inane' can be an answer for 'silly' (I've seen this before). I am not certain of the 'so' bit.

'annie's wrong' is the wordplay.
'wrong' is an anagram indicator.
'annie' with letters rearranged gives 'INANE'.

'to be' is the link.

Can you help me to learn more?

(Other definitions for inane that I've seen before include "Fatuous, vacuous" , "Silly, witless" , "unimaginitive" , "Fatuous, mindless" , "Empty-headed" .)

I've seen this clue in the Evening Standard.

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