Looking at the von Neumann–Bernays–Gödel (NBG) axioms of Set Theory in [Wikipedia](https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory) I noticed that it has two axioms of permutation: one for circular permutations and another for transpositions. 


On the other hand, it is well-known that the group of finitary permutations of $\omega$ is generated by transpositions only. 


So my question: *Can those two permutation axioms in NBG be replaced by a single axiom?* (In order to minimize the number of axioms).


Or at least, can the axiom of transposition:


$\forall A\;\exists B\;\forall x\;\forall y\;\forall z\;((x,y,z)\in B\;\Leftrightarrow\;(x,z,y)\in A)$


be replaced by the simpler one:


$\forall A\;\exists B\;\forall x\;\forall y\;((x,y)\in B\;\Leftrightarrow\;(y,x)\in A)$?




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**Added in Edit.** It seems that two permutation axioms of NBG can be replaced by the following their versions:


The classes $\{((x\_1,x\_2,x\_3),(x\_3,x\_2,x\_1)):x\_1,x\_2,x\_3\in\mathbf V\}$ and 
$\{((x\_1,x\_2,x\_3),(x\_1,x\_3,x\_2)):x\_1,x\_2,x\_3\in \mathbf V\}$ exist. Of course those axioms can be written via formulas, which will have only one existentional quantifier instead of $\forall\mathbf A \exists\mathbf B$ in the original axioms. 




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**Added in Next Edit.** Reading the paper of [Kanamori](https://www.researchgate.net/publication/220366286_Bernays_and_Set_Theory) about Bernays
I discovered (for myself) that the 


**Axiom of Inversion:** for every class $X$ the class $\{(y,x):(x,y)\in X\}$ exists;


and 


**Axiom of Associativity:** for every class $X$ the class $\{((x,y),z):(x,(y,z))\in X\}$ exists;


that appeared in our discussion with Pace Nielsen, have been already included to the list of 20 axioms of Bernays in his 1931 letter to Godel who adopted Bernays' system but replaced the Axioms of Inversion and Associativity by the Axioms of Cyclic Permutation and Transposition. The Bernays axioms modified by Godel were later reproduced in the book of Mendelson who coined the name NBG and popularized this system in his classical textbook on Mathematical Logic. 



In the von Neumann–Bernays–Gödel axiomatic system, the Axiom of Transposition can be simplified
I believe that your intended question was something along the lines of whether or not the standard axioms can be simplified or reduced. This is in the spirit of Exercise 13.4 in Jech's "[Set Theory](https://link.springer.com/book/10.1007%2F978-3-662-22400-7)", where one shows that some of the usual Gödel operations are definable in terms of the others.


So I interpret your questions as asking: Can we replace the two permutation axioms of NBG with a single axiom of the form
$$
\forall A\ \exists B\ \forall x\_1\ \forall x\_2\ \dotsc\ \forall x\_n\ ((x\_1,x\_2,\dotsc, x\_n)\in B \leftrightarrow (x\_{\sigma(1)},x\_{\sigma(2)},\dotsc, x\_{\sigma(n)})\in A)
$$
where $\sigma\in S\_n$ is some fixed permutation on $n$ letters, and $n$ is some fixed (meta) natural number. Or, perhaps more naturally, you would want to assert the existence of the class
$$
\{((x\_1,x\_2,\dotsc, x\_n),(x\_{\sigma(1)},x\_{\sigma(2)},\dotsc, x\_{\sigma(n)}))\ :\ x\_1,x\_2,\dotsc,x\_n\in V\}.
$$
I'm also assuming that you still want to interpret ordered $n$-tuples in the usual way, (meta-)recursively using Kuratowski's trick.


I would guess that the answer to this question is no, but this is clearly not a trivial problem. The paper Gutan and Kisielewicz - "[Rings and semigroups with permutable zero products](https://www.sciencedirect.com/science/article/pii/S0022404905001921)" ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=2235365)) approaches a very similar question; in the context of that paper permutations in $S\_n$ can be considered as permutations in $S\_{n+1}$ in a very concrete way similar to how the permutation axioms of NBG can be lifted from 3-tuples to $n$-tuples. I imagine that the permutation $\sigma$ in the context of NBG induces the same "eventual behavior" as observed in that paper.


(Note: Your parenthetical statement, which reads "to minimize the number of axioms," is nonsense. By taking the conjunction of the finitely many axioms of NBG you now have minimized the number the axioms.)




---


In a sense, one can modify Taras's answer to his own question to get the permutation axiom down to only inversion, with another axiom handling the moving of parentheses.


Throughout, assume the **axiom of inversion**: For each class $X$, the class $X^{-1}=\{(y,x)\, :\, (x,y)\in X\}$ exists. The axiom of inversion will act as our sole permutation axiom.


Consider the new axiom of **associativity of parentheses**: For each class $X$, the class
$$
X^{p}=\{((x,y),z)\, :\, (x,(y,z))\in X\}
$$
exists. In a sense, this axiom will allow us to move parentheses around however we like.


Now, circular permutation $\pi\_3$ follows, since $\pi\_3[X]=(X^{-1})^{p}$. Conversely, associativity of parentheses follows from circular permutation since $X^{p}=\pi\_3[X^{-1}]$. Also notice that the class $(\pi\_3^2[X])^{-1}$ has the parentheses moved to the right instead of the left, so we can shift either direction.


The identity class function $I$ can now be defined, as done by Taras, additionally using axioms of domain, membership (the class $E$ exists), complement, and the existence of $V$.


Now, let me show that $S\_1=\{(((a,b),c),((a,x),y))\,:\, a,b,c,x,y\in V\}$ exists. Taking
$$
S\_1'=\{(((a,b),c),(a,x))\, :\, a,b,c,x\in V\}
$$
we see that $S\_1^p=S\_1'\times V$. So (assuming closure under products from $V$) it suffices to show that $S\_1'$ exists. But $(S\_1')^{p}=S\_1''\times V$ where
$$
S\_1''=\{(((a,b),c),a)\, :\, a,b,c\in V\}
$$
so it suffices to show that this class exists. Now $((S\_{1}'')^{-1})^{p}=S\_1'''\times V$ where
$$
S\_1'''=\{(a,(a,b))\, :\, a,b\in V\}.
$$
Applying associativity, this new class becomes $E\times V$, which exists.


Similar computations, of rearranging coordinates, and peeling off a free coordinate, show that $S\_2$ and $S\_3$ exist, where
$$
S\_2=\{(((a,b),c),((x,y),b))\, :\, a,b,c,x,y\in V\}
$$ 
and
$$
S\_3=\{(((a,b),c),((x,c),y))\, :\, a,b,c,x,y\in V\}.
$$
Now $S\_1\cap S\_2\cap S\_3$ is the class function
$$
\{(((a,b),c),((a,c),b))\, :\, a,b,c\in V\}.
$$
The axiom of transposition (of the last two entries in triples) now follows as in Taras's answer.


