In any topos (or in intuitionistic Zermelo-Fraenkel set theory IZF), the Axiom of Choice implies the law of excluded middle: for every proposition $P$, either $P$ or $\neg P$.

Let $B = \overline{B}(0, 1) \subset \mathbb{R}^3$ be the closed unit ball. There exist pairwise disjoint sets $A_1, A_2, A_3, A_4, A_5 \subset B$ and rigid motions (isometries) $\sigma_1, \sigma_2, \sigma_3, \sigma_4, \sigma_5: \mathbb{R}^3 \to \mathbb{R}^3$ such that \begin{align*} B = A_1 \cup A_2 \cup A_3 \cup A_4 \cup A_5 \end{align*} and \begin{align*} B = \sigma_1(A_1) \cup \sigma_2(A_2) \cup \sigma_3(A_3) = \sigma_4(A_4) \cup \sigma_5(A_5). \end{align*} That is, the five pieces can be rearranged to form two disjoint copies of $B$.

Assuming the Axiom of Choice, every vector space over any field has a basis.

Assuming the Axiom of Choice, there exists a subset $V \subset [0, 1]$ that is not Lebesgue measurable.

Every set can be well-ordered. That is, for every set $S$, there exists a well-ordering $\leq$ on $S$.

Let $(P, \leq)$ be a nonempty partially ordered set in which every totally ordered subset (chain) has an upper bound in $P$. Then $P$ contains at least one maximal element --- that is, an element $m \in P$ such that if $m \leq p$ for some $p \in P$, then $p = m$.

The [set](/page/Set) $\mathbb{R} \setminus \mathbb{A}$ of transcendental numbers is uncountable. In particular, transcendental numbers exist.

Let $\sim$ be an [equivalence relation](/page/Equivalence%20Relation) on a [set](/page/Set) $X$. Then the equivalence classes of $\sim$ form a partition of $X$. Conversely, given a partition of $X$, the relation "$a \sim b$ if and only if $a$ and $b$ lie in the same part" is an equivalence relation whose equivalence classes are precisely the parts of the partition.

Let $A$ and $B$ be [sets](/page/Set). If there exist injections $f: A \to B$ and $g: B \to A$, then there exists a bijection $h: A \to B$.

For any [set](/page/Set) $X$, there is no surjection from $X$ onto $\mathcal{P}(X)$. In particular, there is no bijection between $X$ and its power set $\mathcal{P}(X)$.