Characterization of the span of a set

Question

Denote $[S]$ as the span of $S$.
Let $S$ be a subset of a vector space $V$ and $\alpha \in V$. Then $\alpha \in [S]$ if and only if $[S]=[S\backslash \{\alpha\}]$.

Answer 1

Maybe you meant to conclude "$\alpha\in[S]$ if and only if $[S]=[S\cup\{\alpha\}]$", which is correct. But you cannot expect people to read your mind, so please state more carefully.

For the edited question "$\alpha \in V$. Then $\alpha \in [S]$ if and only if $[S]=[S\backslash \{\alpha\}]$" it is certainly incorrect; for one thing if $\alpha\notin S$ then $[S]=[S\backslash \{\alpha\}]$ holds trivially, regardless of whether $\alpha\in [S]$, while if $\alpha\in S$ then $\alpha \in [S]$ holds trivially, regardless of whether $[S]=[S\backslash \{\alpha\}]$.