I am reading the following proof:
Why can we assume bijection between $x$ and $n\cup \{n\}$ exists in the end of the first paragraph? Isn't this what we trying to prove? Further, in the last paragraph, why do we need to show $x\cap n=n$? How is this statement related with $x=n\cup \{n\}$? And if we already have $x\subseteq n\cup \{n\}$ and $n\in x$, why cannot we directly deduce $x\cap n = n$?