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Why PROP (Set of all propositions) is a set by ZF axioms?

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 ...
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3answers
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Predicate Calculus with Sets - Question about use of an axiom

Greets again StackExchange, I am watching an online lecture, and I believe that my instructor has misused an axiom. Is my concern warranted? $$\begin{align*} \text{Given:}& {P \subseteq (Q \cap ...