In the notes that I am using to study Set Theory, the Axiom of Foundation is presented as follows:
Axiom of Foundation: $$\forall x ( \exists y (y \in x) \rightarrow \exists y ((y \in x) \land \forall z ( (z \in x) \rightarrow \lnot (z \in y)))) $$
According to the textbook, this asserts that bottomless sets are empty. However, I am a bit confused about why that is the case. My interpretation of the statement is that if a set is non-empty, then at least one of its elements does not contain any other elements of the original set.
However, to me, this doesn't feel sufficient to make the claim that bottomless sets are empty, since I'm not really clear on where bottomless sets even feature in this particular formulation of the Axiom of Foundation.
Regarding what exactly we mean by "bottomless", the textbook doesn't define the term before this point, so again, this is potentially another source of my confusion if I am misunderstanding what it is referring to.
I would be grateful for any clarification here.
The definition on Wikipedia is equivalent to the one stated in the question by firstly replacing $\exists y (y \in x)$ with $ x \neq \emptyset$ and secondly replacing $\forall z ((z \in x) \rightarrow \lnot (z \in y ))$ with $y \cap x = \emptyset$ (as these are clearly equivalent sentences).
Now we can consider this formulation and directly employ the proof given on Wikipedia (with adjusted notation).
Which contains further details in the following link.