The quality of having one purpose - to remain unmarried? (10)

'the quality' is the definition.
(singleness is a kind of quality)

'having one purpose to remain unmarried?' is the wordplay.
I cannot really see how this works, but
'one' could be 'single' (similar in meaning) and 'single' is found in the answer.
The remaining letters 'ness' is a valid word which might be clued in a way I don't see.
This may be the basis of the clue (or it may be nonsense).

'of' is the link.

Can you help me to learn more?