I need help with some formal proof writing Okay so I have started to teach myself "Abstract algebra", but before that, I have to clear the preliminaries such as set theory and Linear algebra.
So I was trying to prove the idempotent law
$A \cup A = A$
I need help if someone could check if the proof looks formal or not.
The Proof:
Lemma: $A \cup A = A$
Proof: $\text{Let x be an arbitrary element of} A \cup A, \\\Rightarrow x\in A \cup A\\\Rightarrow x \in A \lor x\in A \\ \Rightarrow x \in A \\ \Rightarrow A \subset A \cup A \hspace{20mm} ... (i) \\.\\\text{Let y be an arbitrary element of A,}\\ \Rightarrow y \in A \\ \Rightarrow y \in A \lor y\in A \\ \Rightarrow y \in A \cup A \\ \Rightarrow A \cup A \subset A \hspace{20mm}...(ii)\\.\\\text{From statement (i) & (ii) we conclude that,}\\ A \cup A = A\\\hspace{40mm}Q.E.D$
Okay so this pretty much, how I did it, please check it and tell me at what places I had to write some extra steps or which steps can be removed and is the proof formal or not?
Thank you for reading!
(PS: I am pretty bad at latex so maybe parts of the proof look bad, also I am a high school student and haven't taken any Proof based or discrete mathematics class.)
 A: This might just be a copy-paste error but in the first part $(i)$ the last line is not the correct conclusion, the preceding would imply $A \cup A \subset A$.
Other than that everything seems fine, whether you should remove or add steps depends on your level and who your audience is. But if this is an early exercise in proof writing it is usually a good idea to include every logical step, as you did.
I would recommend using full sentences instead of just "$\Rightarrow$" symbols, this makes it much easier to read and is usually the better way to do it.
A: Firstly, you mistakenly wrote $A\subset A\cup A$ in the first part where it should be $A\cup A\subset A$. Same with the second part.
Second, the things you need to add/remove depend upon the level. If you want to make the proof a little bit more formal and better, you can remove the $\implies$ sign from where you could use something else which maybe suits better. For example, replacing the second sign by "By the definition of union, we have".

Third, (not very important) you use the notation $\subset$, which is okay as many people still use it. There is an alternative notation $\subseteq$ defined analogously to $\leq$ and $<$.
Also, appreciate your effort of typing context and staying by the policy. It is not something seen every time here.
Hope this helps. Ask anything if not clear :)
