I have been looking at a proof of a technical forcing lemma, and I have a couple of questions.
Here is the setup: $(N, \epsilon) \prec (\mathbf{H}( \chi ), P, \epsilon ) $ is a countable submodel, $P$ is a notion of forcing.
Lemma: The following conditions are equivalent:
1. $q \in P$ is $N$-generic, and
2. $\forall \dot{\alpha} \in N, ( N \models$ "$\dot{\alpha}$ is a $P$-name for an ordinal" $\rightarrow q \Vdash \dot{\alpha} \in N)$
To prove $2 \rightarrow 1$, let $A \in N$, $N \models $"A is a maximal antichain." In $N$, we enumerate $A = \{ a_{\xi} : \xi < \kappa \}$. Then, we define $\dot{\alpha} = \{ ( \xi, a_\xi ) : \xi < \kappa \}$.
The argument then claims that $\dot{\alpha}$ is a $P$-name for an ordinal. But, isn't it actually a $P$-name for a set containing a single ordinal? So how could we apply 2?