The abbreviation ``TFAE'' is shorthand for ``the following are equivalent''. It is used before a set of equivalent conditions (each implies all the others).

In a definition, when one of the conditions is somehow ``better'' (simpler, shorter, ...), it makes sense to phrase the definition with that condition, and mention that the others are equivalent. ``TFAE'' is typically used when none of the conditions can take priority over the others. Actually proving the claimed equivalence must, of course, be done separately.