\begin{definition}[Definition:Matroid Axioms/Axioms 3]
Let $S$ be a finite set.
Let $\mathscr I$ be a set of subsets of $S$.
The '''matroid axioms''' are the conditions on $S$ and $\mathscr I$ in order for the ordered pair $\struct {S, \mathscr I}$ to be a matroid: 
{{begin-axiom}}
{{axiom | n = \text I 1
        | mr= \O \in \mathscr I 
}}
{{axiom | n = \text I 2
        | q = \forall X \in \mathscr I: \forall Y \subseteq S
        | mr= Y \subseteq X \implies Y \in \mathscr I
}}
{{axiom | n = \text I 3''
        | q = \forall U, V \in \mathscr I
        | mr= \size V < \size U \implies \exists Z \subseteq U \setminus V : \paren{V \cup Z \in \mathscr I} \land \paren{\size{V \cup Z} = \size U}
}}
{{end-axiom}}
Category:Definitions/Matroid Theory
\end{definition}