How may I prove that such a statement if set operations are yet to be defined in my course (Introductory Proof-Writing).
My questions:
$1$. Is my proof alright?
$2$ May you give any alternative proofs taking into account we have not defined set operations in my course?
Here's my attempt:
Proof.
We need to show $A=C$ or $$A\subseteq C \wedge C\subseteq A$$
Let's notice that from our hypotheis we have $x\in C \longleftrightarrow x\in A \wedge x\in B$, which is logically equivalent to $$C\subseteq A \wedge C\subseteq B$$
Remains for us to prove $A\subseteq C$ and to that end let $a\in A$.
By way of contradiction suppose that $a\notin C$. This implies $a\notin A \vee a\notin B$.
Because $a\in A$, $a\notin B$. A contradiction to the fact $A\subseteq B$. Thus, the statement is true.