Let $\dagger$ be a binary operation acted on two sets. It has the properties $$\forall A\forall B\forall C \Big(A\dagger B=A\dagger C\iff B=C\Big)$$ and $$\forall A\forall B\Big(A\dagger B=\Big((A\cap B) \dagger (A\cap B)\Big)\dagger(A\Delta B)\Big)$$
Here, I'm using $\Delta$ to represent the symmetric difference: $A\Delta B=(A\cup B)\setminus(A\cap B)$
From the second property, we can infer that $\dagger$ is communitive.
From these properties alone, can we prove that $\dagger$ is associative? I.e; $$\forall A\forall B\forall C \Big((A\dagger B)\dagger C = A\dagger (B\dagger C)\Big)$$
I've tried expanding $(A\dagger B)\dagger C = A\dagger (B\dagger C)$ recursively through property two, but it gets quite messy. I have a hunch that it is associative, but I can't seem to continue.
Edit: I hadn't a specific set theory in mind, but in case certain axioms are aren't naively unnecessary, let's assume the realm of ZFC.
If your curious, I'm afraid it's quite difficult to explain why this came up without boring everyone. But briefly, I've been studying set theories that are highly related to $p$-adic systems, and this operation comes up when defining hyperoperations. Any help is highly appreciated.