Definition 1. Let $V$ be a vector space. $B \subseteq V$ is a basis of $V$ if $B$ spans $V$ and is linearly independent. The length of a basis is its cardinality.
The following three statements are equivalent to $\AC$:
Proposition 1 (Basis Extension). Let $V$ be a vector space. For any linearly independent subset $S \subseteq V$, there is a basis $B\supseteq S$.
Proposition 2 (Basis Selection). Let $V$ be a vector space. For any spanning subset $S \subseteq V$, there is a basis $B \subseteq S$.
Proposition 3 (Existence of Basis). Every vector space has a basis.
Proposition 4 (Steinitz Exchange Lemma). Let $V$ be a vector space. If $A \subseteq V$ is linearly independent and $B \subseteq V$ spans $V$, then there is a set $T \subseteq A$ with $\lvert T \rvert = \lvert B \rvert - \lvert A \rvert$ such that $A \sqcup T$ spans $V$.
Proposition 5. All bases of a vector space have the same length.