I'm going through Enderton's Elements of Set Theory, as I heard it is a gentle introduction to set theory. I'm a little confused on how the subset axioms are used. In the text, the axiom is given as:
Subset Axioms For each formula $\underline{\hspace{1cm}}$ not containing $B$, the following is an axiom: $$ \forall t_1\cdots\forall t_k\forall c\exists B\forall x(x\in B\Leftrightarrow x\in c \wedge \underline{\hspace{1cm}}). $$
Later, the text asserts the existence of the set $\{A\cup X\ | \ X\in\mathscr{B}\}$, call it $\mathscr{D}$, saying that since $A\cup X\subseteq A\cup\bigcup\mathscr{B}$, $\mathscr{D}$ is a subset of $\mathscr{P}(A\cup\bigcup\mathscr{B})$. A subset axiom produces $$ \{t\in\mathscr{P}(A\cup\bigcup\mathscr{B})\ |\ t=A\cup X\ \text{for some}\ X\in\mathscr{B}\}. $$
My question is, what exactly is that subset axiom? Would it be something of the form $$ \forall t_1\cdots\forall t_k\forall c\exists B\forall x(x\in B\Leftrightarrow x\in c \wedge [(x=t_1\cup t_2)\vee(x=t_1\cup t_3)\vee\cdots\vee(x=t_1\cup t_k)]) $$
where we take $A$ to be a particular instance of $t_1$, $\mathscr{P}(A\cup\bigcup\mathscr{B})$ to be a particular instance of $c$, and $t_2,\dots,t_k$ to be particular instances of members of $\mathscr{B}$?
Or perhaps something like $$ \forall t_1\forall t_2\forall t_3\forall c\exists B\forall x(x\in B\Leftrightarrow x\in c \wedge [\exists t_3\in t_2(x=t_1\cup t_3)]) $$
where in this case $\mathscr{B}$ is a particular instance of $t_2$?, and then $\mathscr{D}$ would be the set $B$ which has been proven to exist. Thanks, right now I'm just not used to what should go in that blank spot for the formula.