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", 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" (MSN) 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.)

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", 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" (MSN) 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.

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", 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 $$ \forall A\ \exists B\ \forall x_1\ \forall x_2\ \ldots\ \forall x_n\ ((x_1,x_2,\ldots, x_n)\in B \leftrightarrow (x_{\sigma(1)},x_{\sigma(2)},\ldots, x_{\sigma(n)})\in A) $$ 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 $$ \{((x_1,x_2,\ldots, x_n),(x_{\sigma(1)},x_{\sigma(2)},\ldots, x_{\sigma(n)}))\ :\ x_1,x_2,\ldots,x_n\in V\}. $$ 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 "Rings and semigroups with permutable zero products" 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.)

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", 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" (MSN) 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.)