What are the requirements for the conditioning probabilities in the derivation of Bayes' theorem?

The derivation of Bayes' theorem requires that the conditioning probabilities be nonzero. Specifically, when deriving

P(A|B)

from the intersection, we divide by

P(B)

, so

P(B)

must be positive. Similarly, the definition of

P(B|A)

requires

P(A)

to be nonzero.

Conditions

Recall the definition of conditional probability: $P(A|B) = P(A \cap B) / P(B)$ .
Note that division by zero is undefined, so $P(B)$ must be nonzero.
Recall the definition for the reverse conditional: $P(B|A) = P(A \cap B) / P(A)$ .
Note that $P(A)$ must also be nonzero.
Conclude that both conditioning probabilities must be strictly positive for the standard derivation to hold.

The script states: 'The two conditional definitions require nonzero conditioning probabilities.'

Assuming Bayes' theorem applies when the evidence has zero probability.
Forgetting that the prior probability of the event of interest must also be nonzero for the reverse conditional to be defined.
Thinking that $P(A \cap B)$ being nonzero is sufficient without checking the denominators.