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Mum's one old bird without energy to join rave (9)

Ross

I believe the answer is:

deodorant

I'm a little stuck... Click here to teach me more about this clue!

'mum's one' is the definition.
Although both the answer and definition are singular nouns, I cannot understand how they can define each other.

'old bird without energy to join rave' is the wordplay.
'old bird' becomes 'dodo' (extinct bird).
'without' means one lot of letters goes inside another ('without' can be similar in meaning to 'outside').
'energy' becomes 'e' (physics symbol as in E=mc2).
'to join' is a charade indicator (letters next to each other).
'rave' becomes 'rant' (synonyms).
'dodo' placed around 'e' is 'deodo'.
'deodo'+'rant'='DEODORANT'

Can you help me to learn more?

(Other definitions for deodorant that I've seen before include "Underarm protection?" , "Remover of smells" , "for keeping smells away" , "Toiletry product" , "Underarm spray, perhaps" .)

I've seen this clue in The Independent.
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