I have a possibly silly question about the Drinfeld upper half plane. It is "well-know" that if $K$ is a complete local field then $\Omega_K = \mathbb{P}^1(\mathbb{C}_k) \backslash \mathbb{P}^1(K)$ has the structure of a rigid analytic space. A proof of this is given in proposition 6.1 of Drinfeld's paper "Elliptic Modules I" (1974). From what I can understand it goes as follows.
We know $\mathbb{P}^1_K$ has the structure of a rigid analytic space. First we define an increasing chain of admissible open affinoids $U_1 \subset U_2 \subset \cdots$ in $\mathbb{P}^1_K$ whose union is $\Omega$. Drinfeld then states that to show $\Omega$ is an admissible open subset it suffices that:
For any morphism from an affinoid space $f: \mathrm{Sp}(A) \to \mathbb{P}^1_K$ factoring through $\Omega$, $f$ in fact factors through some $U_i$.
This seems like a natural enough claim. In fact, it's not too hard to show that for a general rigid analytic space $X$, if $\{ U_i \}$ is a set-theoretic covering by admissible (not necessarily affinoid) opens, then this covering is admissible if and only if for all morphisms $g : \mathrm{Sp}(A) \to X$, the cover $\mathrm{Sp}(A) = \cup_i g^{-1}(U_i)$ has a finite affinoid refinement.
In particular, if we knew $\Omega$ was an admissible open, the condition in the box above would imply that $\{ U_i \}$ is an admissible cover of $\Omega$.
My question is: what am I missing? How do we know that $\Omega$ is an admissible open?
I know that we can pick some admissible covering of $\mathbb{P}^1_K$ by affinoid analytic spaces, say $\mathbb{P}^1_K = \cup_j V_j$ with $V_j \cong \mathrm{Sp}(A_j)$. Then $\Omega$ is an admissible open $\iff \Omega \cap V_j$ is admissible for all $j$. I feel like this should help.
As mentioned in the question, it's enough to show $\Omega \cap V_j$ is admissible open in $\mathbb{P}^1$ for all $j$. Clearly $\Omega \cap V_j = \cup_{i} \left( U_i \cap V_j \right)$ is a union of admissible opens (in $V_j$). Let $W_{ij} = U_i \cap V_j$. Each $W_{ij}$ is admissible in both $V_j$ (when considered as a rigid space) and $\mathbb{P}^1$. Furthermore, by definition, if we know $\cup_i W_{ij}$ is admissible in $V_j$ then it's also admissible in $\mathbb{P}^1$.
Suppose we have a morphism $f : \mathrm{Sp}(B) \to V_j$ with image contained in $\Omega \cap V_j$. Composing with the natural embedding $g_j : V_j \to \mathbb{P}^1$, we get a morphism as in the box in the question. Applying the claim in the box, we see that it factors through some $U_{i_0}$, which implies $f$ factors through $W_{i_0 j}$, so $\{ f^{-1}(W_{i_0 j}) \}$ is a (single-element) finite cover of $\mathrm{Sp}(B)$. This is exactly what we needed to show $\cup_i W_{ij}$ is admissible open in $V_j$ so we're done.