Let A and B be countable infinite sets. Being both countable, a one-to-one correspondence between the set’s elements can be established. A new correspondence can also be established between A and the union of all elements in A and B, since this is another countable set.
Intuitively, it would seem even more obvious that the same would apply if A was uncountable. The larger set would still be uncountable. However, the countability is part of the proof in the first case, such as two car lanes merging into one. The same technique is not available in the second case.
(I note the question: An uncountable set minus a countable set is still uncountable, but is that equivalent to my question? - It would be if there is a one-to-one correspondence between all uncountable sets, but I don't think this is obvious)
How is such a proof delivered?