Is every element in a set set? In a set model it is obvious true. However in the class universe it is another story since it need to be shown not a proper class.
Let $A$ be a set, $a \in A$.
Firstly consider about the Axioms of Comprehension. Of course $a= \{x \in \cup A|x \in a\}$. However $x \in a$ may not a formula in the language of sets since $a$ may not a constant symbol. But if we can find a formula $\varphi_a(x)$ in the language such that $\varphi_a(x) \iff x \in a$ then the Axioms of Comprehension can be spelt. Unfortunately this kind formula does not always exist if $A$ is uncountable whereas the language is countable.
Then the Axioms of Replacement. If $a=\emptyset$ it is obviously a set; else it contains at least one element, let $m$ be one of them. The being $f=\{(x,y)|x\in \cup A \land (x\in a \rightarrow y=x)\land(x \not \in a \rightarrow y=m)\}$, if it is a function then $a$ is a set by Axioms of Replacement. But note that $a$ and $m$ may not definable in the language hence there may no formula $\psi(x,y)$ to define $f$.