Taking place in ZFC.
The axiom of class construction (ACC) states that given a statement expressed in the accepted terms of set theory and a collection of elements $x$, there exists a class of elements which satisfies the statement $P(x)$.
I am tasked with proving that the class $\{A, A\}$ is equivalent to just $\{A\}$. My intuition is to say that since each collection above of $A$ are classes that satisfy the statement $P(x)$, they are the same class.
My hang up is that this would seem to require the axiom to imply every class can be created from the rules stated in the ACC. ACC says there exists a class to satisfy any statement, but it seems like there is room to say that there exists classes that don't necessarily fulfill a property. Am I misreading the axiom (does it actually imply this and I don't see it?). Generally, am I on the right track for proving this?
Axiom 1 of ZFC, axiom of extent, according to this book says that if A and B are classes and:
[ $x \in B \leftarrow \rightarrow x \in A$ ]$\leftarrow \rightarrow$ [ $A = B$ ]
So the doubleton {D, D}, and the singleton {D} can be replaced for $A$ and $B$ in the statement above to show that they are equal according to A1.