I'm missing something about transfinite arithmetic using Fundamental theorem of arithmetic

Question

Ok, So I know this proof is wrong, but I can't find the error.

1. The fundamental theorem of arithmetic says that every natural number can be uniquely expressed as a product of primes. I usually see this explained with multistep. (Definition)
2. So the naturals are bijective to the set of multisets of primes (Right? [This is the wrong step]) (From Step 1)
3. Primes are countable and thus bijective to naturals (Definition)
4. The set of multisets of primes is bijective to the set of multisets of naturals (This seems like the error, but I don't see why) (From step 3)
5. So the naturals are bijective to the set of multisets of naturals (At this point I know I'm wrong) (From steps 2 and 4)
6. The powerset of naturals is a proper subset of the set of multisets of naturals (Definition)
7. The reals are bijective to the powerset of naturals (outside theorem)
8. The reals are a proper subset of the naturals (from steps 5,6,7)
9. 22/7 is real and not natural (Definition)
10. The reals are not a proper subset of the naturals (From Step 9)
11. Contradiction. (From steps 8, 10)

What am I doing wrong here?