This principle as written isn't appropriate for the class of theories including NF and NFU. If the formula $\phi$ is restricted to be stratified and have $x$ and $y$ of the same relative type then the principle is true in NF and so in any of its fragments (this is the real answer).

In NF, the universe is a set, the set of all singletons is a set, and there can be no bijection between them. Suppose there were such a bijection f from singletons onto sets. Then we could define the set $R = \{x : \lnot (x \,E\, f(\{x\}))\}$ (this definition would be stratified). Now consider  $f^{-1}(R) = \{r\}$. $r \,E\, R$ iff $\lnot r \,E \,R$ follows.

There are fragments of NF in which this argument does not work: these inevitably have seriously impaired comprehension principles and are very weak. I believe that in versions of NF with predicativity restrictions on comprehension, one can arrange for all infinite sets to be the same size. But mathematically these systems are quite weak, and the principle Zuhair suggests becomes true in a trivial sense.

This principle as written isn't appropriate for the class of theories including NF and NFU. If the formula $\phi$ is restricted to be stratified and have $x$ and $y$ of the same relative type then the principle is true in NF and so in any of its fragments (this is the real answer).

In NF, the universe is a set, the set of all singletons is a set, and there can be no bijection between them. Suppose there were such a bijection f from singletons onto sets. Then we could define the set $R = \{x : \lnot (x \,E\, f(\{x\}))\}$ (this definition would be stratified). Now consider  $f^{-1}(R) = \{r\}$, $r \,E\, R \iff \lnot \ r \,E \,R$ follows.

There are fragments of NF in which this argument does not work: these inevitably have seriously impaired comprehension principles and are very weak. I believe that in versions of NF with predicativity restrictions on comprehension, one can arrange for all infinite sets to be the same size. But mathematically these systems are quite weak, and the principle Zuhair suggests becomes true in a trivial sense.

This principle as written isn't appropriate for the class of theories including NF and NFU. If the formula phi is restricted to be stratified and have x and y of the same relative type then the principle is true in NF and so in any of its fragments (this is the real answer).

In NF, the universe is a set, the set of all singletons is a set, and there can be no bijection between them. Suppose there were such a bijection f from singletons onto sets. Then we could define the set R = {x : ~ x E f({x}}} (this definition would be stratified). Now consider f^{-1}(R) = {r}. r E R iff ~r E R follows.

There are fragments of NF in which this argument does not work: these inevitably have seriously impaired comprehension principles and are very weak. I believe that in versions of NF with predicativity restrictions on comprehension, one can arrange for all infinite sets to be the same size. But mathematically these systems are quite weak, and the principle Zuhair suggests becomes true in a trivial sense.

This principle as written isn't appropriate for the class of theories including NF and NFU. If the formula $\phi$ is restricted to be stratified and have $x$ and $y$ of the same relative type then the principle is true in NF and so in any of its fragments (this is the real answer).

In NF, the universe is a set, the set of all singletons is a set, and there can be no bijection between them. Suppose there were such a bijection f from singletons onto sets. Then we could define the set $R = \{x : \lnot (x \,E\, f(\{x\}))\}$ (this definition would be stratified). Now consider $f^{-1}(R) = \{r\}$. $r \,E\, R$ iff $\lnot r \,E \,R$ follows.

There are fragments of NF in which this argument does not work: these inevitably have seriously impaired comprehension principles and are very weak. I believe that in versions of NF with predicativity restrictions on comprehension, one can arrange for all infinite sets to be the same size. But mathematically these systems are quite weak, and the principle Zuhair suggests becomes true in a trivial sense.