$\mathcal M\,$ is a nonempty subset of the set of all subsets of the finite set $X$, such that:
$(1)\quad\; \neg\exists U,V\in\mathcal M:U\subset V\quad$
$(2)\quad\; A\subseteq B\in\mathcal M\wedge C\in\mathcal M\Rightarrow \exists U\subseteq C:A\cup U\in\mathcal M$, alternatively
If $\;\mathcal I=\{U\subseteq X|\exists V\in\mathcal M:U\subseteq V\}$, is $(X,\mathcal I)$ a matroid then?
And reverse, if $\mathcal M$ is the set of all maximal independent sets in $\mathcal I$, for any matroid $(X,\mathcal I)$ does $\mathcal M$ satisfies $(1)$ and $(2)$?