I'm studying logic and sets and I have to say there's a strong similarity between the two. Most boolean/logic axioms also apply to sets. At the end of my course I also studied first-order logic (or predicate logic) and how one can actually define statements using first-order logic. This was quite a revelation to me because it is quite easy to prove first-order logic statements compared to set propositions (my brain just works better at decomposing first-order logic propositions). So I'm wondering, is it possible to prove set propositions using first-order logic? Here's an example of a set proposition I have to prove:
if C ⊆ B then A ⇒ C ⊆ A ⇒ B
(not that A ⇒ B is assumed to mean ¬A ∪ B)
Now I was thinking I could convert this into a first-order logic statement, such as:
Subset(C, B) ⇒ Subset(A ⇒ C, A ⇒ B)
As you can see however I can't seem to understand how to prove that in first-order logic. Anyway, perhaps I am confusing the two and this is not possible but I was wondering what you guys thought about proving these set statements using first-order logic.