In $\sf NF(U)$ it is known that the singleton map $(x \mapsto \{x\})$ is not a set, which is a source of a lot of extremely counter-intuitive results in $\sf NF(U)$. However, a weakening of the criterion of stratification after predicativity that yields the theory $\sf NFP$ is compatible with having models in which all sets are countable. So Cantor's uncountability argument fails in it, and powersets are no longer larger than the set of singleton sets in them. Hence my question:

Is it consistent to add the singleton map to $\sf NFP$? Formally, this is: $$ \sf NFP + \{(x,\{x\}) \mid x=x\} \text { exists} $$.

In $\sf NF(U)$ it is known that the singleton map $(x \mapsto \{x\})$ is not a set, which is a source of a lot of extremely counter-intuitive results in $\sf NF(U)$. However, a weakening of the criterion of stratification after predicativity that yields the theory $\sf NFP$ is compatible with having models in which all sets are countable. So Cantor's uncountability argument fails in it, and powersets are no longer larger than the set of singleton sets in them. Hence my question:

Is it consistent to add the singleton map to $\sf NFP$? Formally, this is: $$ \sf NFP + \{(x,\{x\}) \mid x=x\} \text { exists} $$.