It looks like you are confusing a property of the material conditional—namely, that every material conditional with a false antecedent is a true conditional—with the principle of explosion—that from a contradiction one may (validly) infer anything. Having a single false belief (or even a bunch of them) and having beliefs closed under logical entailment does not necessarily lead to explosion. The set of beliefs has to include a logical contradiction for explosion to occur.
It looks like you are confusing a property of the material conditional—namely, that every material conditional with a false antecedent is a true conditional—with the principle of explosion—that from a contradiction one may (validly) infer anything. Having a single false belief (or even a bunch of them) and having beliefs closed under logical entailment does not necessarily lead to explosion. The set of beliefs has to include a logical contradiction for explosion to occur.
You’re right, I was very sloppy—I’m mainly prepping for the “Relevance logics” post, which rejects the material conditional as we normally use it.
Have corrected it, thanks.