TITLE: How can I be sure that two given sets A and B are truly equal sets?
QUESTION [5 upvotes]: Let's say we have given two sets: $A$ and $B$.
I thought about two possible statements:
1st: $A = B$ iff $|A| = |B|$ and $(\forall x \in A: x \in B)$ and $(\forall y \in B: y \in A)$
2nd: $A = B$ iff $|A \setminus (A \cap B)| = 0$ and $|B \setminus (A \cap B)| = 0$
Now I am curious if this statements are true and how I can prove them (or are they already fundamental/trivial).
Also are there other possible true statements?

REPLY [9 votes]: Obviously $|A| = |B|$ is redundant because of $(\forall x \in A: x \in B) \Rightarrow A \subseteq B$ and $(\forall x \in B: x \in A) \implies B \subseteq A$, therefore $\implies$ $A \subseteq B  \land B\subseteq A \iff A=B$

REPLY [7 votes]: I would just keep it on:$$A=B\text{ if and only if }\forall x\in A[x\in B]\wedge\forall x\in B[x\in A]$$which is the axiom of extensionality.
The demand $|A|=|B|$ is redundant.
The second statement is correct, but rests on cardinality. Let's stay close to the basics.