Formally, this may be expressed as follows:
∀
X
[
∅
∉
X
⟹
∃
f
:
X
→
⋃
X
∀
A
∈
X
(
f
(
A
)
∈
A
)
]
.
{\displaystyle \forall X\left[\varnothing \notin X\implies \exists f\colon X\rightarrow \bigcup X\quad \forall A\in X\,(f(A)\in A)\right]\,.}
Thus, the negation of the axiom of choice states that there exists a collection of nonempty sets that has no