ok, this direction seems ok

i don't like this proof, it doesn't seem valid. it all starts from the red line. how does that follow from anything you've said. the line directly below that makes no sense either. it's just a faulty proof.Suppose A U B = ∅

Case 1: x is in A

x is in A, x is not in B

Therefore A U B <--> ∅

[(for all x in (A U B), x is in ∅) And (for all x in ∅, x is in (A U B)]

Since x is in a, in order for A U B = ∅

A = ∅

Case 2: x is in b

etc.

for the second direction, use the contrapositive. assume either A or B is not empty, then show that this means AUB is not empty either, pretty easy