It is a long standing open problem whether or not the Axiom of Countable Choice implies the fragment of Martin's Axiom either in or in. In this direction, we provide a partial answer by establishing that the Boolean Prime Ideal Theorem in conjunction with the Countable Union Theorem does not imply restricted to complete Boolean algebras in. Furthermore, we prove that the latter (formally) weaker form of and the Δ‐system Lemma are independent of each other in.We also answer open questions (...) from Tachtsis [16] which concern the status of restricted to complete Boolean algebras in certain Fraenkel–Mostowski permutation models of and we strengthen some results from the above paper. (shrink)