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$\exists X (\forall Y (\neg(Y \in X)))$

is whats given in my lecture, but I was wondering, is it the same as

$\exists X (\forall Y (Y \notin X))$

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    @DanChristensen You might be interested in the development of the Scott-Potter axiomatization in Potter's *Set Theory and its Philosophy* which also develops quite a bit before introducing the existential axiom that there *is* a level (i.e. a set which is a partial universe in the hierarchy).2012-10-01

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Yes, this is what $\notin$ means. $A \notin B$ is a shorthand for $\neg(A \in B)$.