In the following let $\mathbf{V} = \bigcup_{\alpha \in \mathbf{ON}} V_\alpha$ denote the cumulative hierarchy. Let $\{\varphi_0, \dots, \varphi_n, \dots \}$ denote a list of all $ZF$ axioms. I am reading the following sentence:
"Given $n \in \omega$, the symbols "$\mathbf{V} \models \{\varphi_0, \dots, \varphi_n\}$" and "$\varphi_0 \land \dots \land \varphi_n$" stand for exactly the same thing." (Just/Weese, p 192)
I don't understand how this is true. The first expression says that $\varphi_i$ are all true in $\mathbf{V}$ given any valuation. The second expression seems to be a formula that may or may not be true but there is nothing saying that it is true in $\mathbf{V}$. Thank you for shedding light into my confusion.