Concise list of ZFC Axioms for beginners I am teaching a course and want to provide students with a simple explanation of the ZFC axioms without technical jargon. I try to define most of the primitive words in the list with the following intro:
An initial naïve approach towards the foundations of Mathematics one considers collections or containers, which are called sets which may contain items called elements. These sets have no duplicate elements and do not a-priori possess any internal structure such as order or size.
The only relation between sets and elements is derived from the logical concept of being in a set, or being an element of a set. 
It is not of interest to know the exact ontological status of these sets or elements, only how identify and how to operate with them. Therefore we take the following statements as true:


*

*There exist a set without elements.

*Two sets with the same elements are identical.

*For any two sets there exists a set which contains them as its elements.

*For any set there exists another set which has contains the elements of the elements of the first.

*For any set there exists another set which contains all of its subsets.

*For any set and any first-order property $p$, there exist a set whose elements satisfy $p$. 

*There exists the infinite set of natural numbers.

*The image of a set under a function is also a set.

*The in relation of belonging is well-founded, this means there is no circularity chain (finite or infinite) such as $A$ is in $B$ and $B$ is in $A$.

*Given two sets, there exists a choice set which has exactly one element of each element of  said sets.
The ordering of the axioms is immaterial, also they are not independent. Initially this appears worrying but in reality this is an infinite list of axioms, since (6, 8) are axiomatic 'schemas', one action per property or function.
I want to know if there is something that needs correction or merits improvement.
 A: Your $(6)$ is misstated: it should say that for any set $A$ and first-order property $p$ there is a set whose elements are precisely the elements of $A$ that satisfy $p$.
Your $(8)$ is conflates functions (i.e., sets of a certain kind) with what might loosely be described as first-order formulas that behave like functions. The replacement schema isn’t needed to get that $f[A]$ is a set if $f$ is a genuine function and $A$ is a set.
Your $(10)$ as stated is unnecessarily confusing. If you want the axiom of choice, you should start with a single set $A$ of non-empty sets and say that there is a function with domain $A$ that picks out an element of each element of $A$.
A: For what it's worth, here's  a student's two cents:
If the motivation is pedagogical, I'd suggest adding a sentence emphasising that elements are sets, and I'd state the axioms as:

A set is empty if and only if has no elements; otherwise, it is nonempty.

*

*There's an empty set.


*Any two sets with all of the same elements are identical.


*For any two sets there's a set which has both of them as its only elements.


*For any set there's another set which has as its elements all and only the elements of each element of the first set.
A set is said to be a subset of another whenever every element of the first set is an element of the second set.


*For any set there's another set which has as its elements all and only the subsets of the first set.


*For any set and any first-order property there's a set whose elements are all and only the elements of the first set that satisfy the property.


*There's a set that has as every natural number as an element.


*For any set the image of that set under a function is also a set.
Two sets are said to be disjoint if and only if they have no element in common.


*If a set has any elements at all it has an element that is disjoint from it.


*If every element of a nonempty set is nonempty there's a set which has as its elements one element chosen from each element of the first set.


*

*I don't consider 'empty', 'nonempty', 'subset', and 'disjoint' to be technical jargon and I think those words are used enough in online (or otherwise public) resources on set theory that they should at least get introductions in class.


*(6) and (8) are axiom schemas and thus really hard to make sense of without technical jargon, so I'll leave that task to the big kids.


*I think 'in relation' needs to be replaced with either 'elementhood relation', 'membership relation', or '$\in$-relation'.


*Despite what was said in the comments, I don't think you need to say anything in particular about natural numbers, and you especially don't need to bring out more axioms because of them. People know what natural numbers are, but to be safe you can just write '$0,1,2,3,...$' on the board when you say it and they'll know what you mean. Besides, just as some presentations include $\varnothing$ as a constant in the language, you could just as easily also include $1,2,3,$ and so on as constants in the language (either initially or by definatorial extension), so it doesn't interfere with conceptualising an infinite set.


*I can imagine getting heat for saying this, but some of your students might not be indoctrinated in mathematics yet, and outside of mathematics it is much more common to say 'there is/are' instead of 'there exists/exist', so if you claim to not be interested in the ontological status of sets it might be a better idea to use the terminology that is more familiar, more concise, and less ontologically suggestive for students. If you also want to be more philosophically conscious/neutral outside of the classroom, maybe you get what I mean when I say Sherlock Holmes doesn't exist, but he is one of the most iconic detectives of all time!
Take what helps and leave what doesn't!
