If I have a set $S$ defined as the smallest set $S$ over an alphabet $A=\left\{ \star, \urcorner,(,), a_0,a_1, \dots \right\}$ ( $S\subseteq \cup_{k \in \mathbb{N}} A^k$) satisfying:
$\bullet \ a_0, a_1, \ldots$ $\in S$
$\bullet $ if $\alpha \in S$ then $ ( \urcorner \alpha) \in S$,
how can I then show for example that $( \alpha \urcorner )$ or $\alpha ()$ is not in the recursively defined set $S$, where $\alpha$ is some element of $S$ (or even $S\subseteq \cup_{k \in \mathbb{N}} A^k$, for greater generality). Of course I can "see" that this is the case, but how can I prove it ?
Maybe because it seems so easy to see that strings like the above can't be in $S$ I can't up with a proof. (Of course, if $\star \alpha$ were in $S$, intuitively it would be clear, that $S$ woulsn't be the smallest set anymore, but how would I then prove this ?)