Looking at the von Neumann–Bernays–Gödel (NBG) axioms of Set Theory in Wikipedia 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:
$\forall A\;\exists B\;\forall x\;\forall y\;((x,y)\in B\;\Leftrightarrow\;(y,x)\in A)$?
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.
Added in Next Edit. Reading the paper of Kanamori 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\};$$\{((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.
