Axiom of choice condition. $\{E_\alpha\}_{\alpha\in A}\ ,\ E_\alpha\subset E\subset\mathbb{R}$ where $A$ is an index set (may be uncountable)
is a condition for the axiom of choice in my lecture notes, along with $E_\alpha\cap E_\beta=\emptyset$ if $\alpha\ne\beta$
Why can't we simply write $\{E_\alpha\}_{\alpha\in A}\ ,\ E_\alpha\subset\mathbb{R}$ instead? Cause for any collection of $E_\alpha$'s their union contains them all and is a subset of $\mathbb{R}$
Just for context, our version of the axiom of choice concludes by saying that $\exists V$ that contains one and only one element from each $E_\alpha$

If you ditch pairwise disjointness, there are counterexamples: Consider $E_1 = \{1\}, E_2 = \{2\}, E_3=\{1,2\}$. Then any choice set $V$ for $\{E_1,E_2,E_3 \}$ must contain $1$ and $2$ but then $V \cap E_3$ contains two elements.
