Some problems are that the question is (1)incompletely stated, and (2)includes unwarranted presumptions:
how can I prove that it can't be done in 7 comparisons?
On the first point, it is unclear what the domain of the sorting problem is: is it random real numbers (with some specified or unspecified distribution), or integers (in some specified or unspecified range), or some other type of objects; are the values guaranteed to be distinct, or could there be duplicates (triplicates, etc.), and so on? If there are duplicates, etc., does it matter whether the partial ordering of items which compare equal is maintained (a.k.a. stable sorting)? None of these things are specified, and they all affect possible (and/or inapplicable) potential solutions to the sorting problem which is behind the stated question. It's also unclear whether the question is about best-case, average-case, or worst-case comparisons (under some specified conditions for some problem domain). And it's unclear whether anything other than comparisons (e.g. amount of data movement, partial-order stability as mentioned above, etc.) is a consideration.
On the second point, one of the responses notes Demuth's 7 comparison solution to non-stable sorting, so the question as stated is an impossible quest; one cannot in fact (correctly) prove the impossibility of something which is in fact possible.
So the short answer to the question as stated is "you can't" (in part because there is a 7 comparison solution to sorting 5 elements with certain constraints (and lacking others), and partly because the question and background do not specify any relevant constraints, details of the problem domain, etc.).