In Propositional Logic when we define the set of all propositions inductively how we can prove such a set(smallest with such properties) does exists? means that the set (of all sets with these properties) under intersection operation is not empty?
Definition 1.1.2 from Van Dalen book:
The Set $PROP$ of propositions is the smallest set $X$ with the properties:
(i) $p_{i}\in X (i\in \mathbb{N})$, $\bot \in X$
(ii) $A,B \in X$ then $(A\wedge B), (A\vee B), (A\rightarrow B), (\neg A) \in X$
I know in propositional logic we model it mathematically based on ZF(C) as a common accepted foundation for mathematics and I know there is another definition by formation sequences, I think it shows at least one such set exists but I want to know without it, how we can show that $PROP$ is a set in ZFC?