Does wealth that prevents receiving Zakat necessarily equate to wealth that obligates paying Zakat?

Primary text

The definition of wealth that prevents receiving Zakat differs from the wealth that mandates paying Zakat, and also differs from wealth that prohibits begging. The individual who has enough to prevent begging but still needs the Zakat funds is distinguished from the person discussed who is not in need of the funds, therefore the latter is prohibited from receiving the charity.