I have a question about a passage of Just/Weese. In the following, let $\mathcal M = \langle M, E \rangle$ be a model of ZFC. Here is the first half of what I'm about to ask a question:
So I asked myself the question of whether the other way around is also true: does every element $a \in M$ correspond to a subset $A \subset M$? Given that the argument for the other direction is purely based on the size of the sets one would expect that yes. But since I constructed a counter example, I think the answer is no (unless my counter example is incorrect (please correct me)):
Let $M = \{ \{\varnothing\}, \{\varnothing, \{\varnothing\}\}, \{\{\varnothing\}\} \}$, $E = \langle \{\varnothing\}, \{\varnothing, \{\varnothing\}\}\rangle, \langle \{\{\varnothing\}\}, \{\varnothing, \{\varnothing\}\}\rangle \}$.
Then $A = \{ \{\varnothing\}, \{\{\varnothing\}\} \}$ corresponds to $a = \{\varnothing, \{\varnothing\}\}$ But the element $a=\{\varnothing\}$ does not correspond to any subset $A \subset M$.
Now look at this:
In particular, "...Let $A$ be the subset of $M$ that corresponds to $a$. ...". But there doesn't necessarily have to be such an $A$, so this passage is not right.
What am I missing? Thanks for your help.
Corresponds is given as follows: