Set Theory Axiomatics

Question

Thinking about Modal Logic made me wonder about Euclidean relations in general, and how they might show up in other areas of math, like Set Theory. I began to think of some examples of Euclidean relations between familiar sets:

1. $\emptyset \in$ {$\emptyset$}, $\emptyset$ $\in$ {$\emptyset$, {$\emptyset$}}, {$\emptyset$} $\in$ {$\emptyset$, {$\emptyset$}},

So, I began to wonder for which sets does this property hold, and given that, what adding this axiom:

$\forall x \forall y \forall z ((x \in y \land x \in z \land y \not =z) \to (y \in z \lor z \in y))$

does to Set Theory in general, and to specific set theories like ZF(C). Is something like this already forbidden by ZF, like the existence of sets that contain themselves?

Answer 1

The axiom is blatantly false, for example consider $x=\varnothing, y=\{\varnothing\}, z=\{\varnothing, 8\}$.

Answer 2

As stated by Shinrin-Yoku, your axiom is -as far as usual set theory goes- not true.

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I'll answer the question:

$$ \begin{split} \textbf{Is something like this already forbidden by ZF,}\\ \textbf{like the existence of sets that contain themselves?} \end{split}$$

There is; the Axiom of Regularity guarantees no set is a member of itself.