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You know, this is probably not the answer everyone is looking for but I believe that there is a trichotomy axiom for the cardinality of sets, much like for the real numbers.
So, practically by definition, if |A|=<|B| and |B|=<|A|, then |A|=|B|. I use =< to not confuse it with the implication sign.
Anyway, in case you're unfamilar with the trichotomy axiom, it states that any two elements, x and y, must have exactly one of the following relationships: x < y, x = y, or x > y. This cannot be proven for the real numbers and, instead, is taken as an axiom. Perhaps the same can be said for set cardinality?
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