Leaving aside the fact that talking "informally" about infinities easily leads to wrong conclusions, the OP wasn't even talking about a strict subset relationship. The comparison was between a set of proofs and a set of facts, which are sets of different mathematical objects. So at best you can say that each proof proves a fact (a fact can have several proofs) and the set of facts that have a proof is a strict subset of all facts. Or, informally, there are more facts than facts which can be proved. Oh, wait, this just restates the theorem we're trying to prove in the first place and doesn't give any insight into why it's correct.

Yes, but the use of "which in turn" suggests that was not meant; it suggests that what follows is a real deduction and not a mere rephrasing.

And if you read about Cantor you will find he defined a term called 'countably infinite'. No 'countably infinite' set has a higher cardinality than any other.