In class universe, on what basis can a conclusion that every element of a set is set be made? 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 $f$. 
 A: In the universe of sets all the objects are sets. In particular the statement $x\in y$ is saying that $x$ and $y$ are sets and the relation $\in$ as interpreted by the universe agrees that $x \in y$ is true (note that in this case $\in$ is used as a syntactic object whereas in the first use above it meant as a semantic object).
Therefore if the universe knows that $x\in y$ at all then it knows about $x$, therefore $x$ is a set.
Ah, but you could retort with the following question "But why can we write $\alpha\in\mathsf{Ord}$ if the ordinals do not form a set?", and indeed this is the right question to ask. The answer is that when we write that we actually write that $\varphi(\alpha)$ is true, where $\varphi$ is the formula describing the class of ordinals (i.e. transitive and well-ordered by $\in$). This is a step in the proof so we simply use the fact that $\exists x\varphi(x)$ and then we take $\alpha$ to be an instance of such $x$.
There is no semantic object corresponding to the ordinals, though. At least in ZFC. If one talks about class-set theories (e.g. NBG or MK) then this is different story. In those theories it is common to define sets as elements of classes anyway, so the question itself becomes trivial.
