Let's say that a formula in the language of set theory is flexibly stratified iff there exists a function $f$ from variable symbols to $\omega$ such that if $x=y$ appears in the formula, then $f(x)=f(y)$, and if $x\in y$ appears in the formula, then $f(x)<f(y)$ (this is in contrast to regular stratification which requires $f(y) = f(x) + 1$ exactly).

I'm going to define Flexible NF as:

• Extensionality (same as in NF)
• Flexibly Stratified Comprehension (same as in NF except allowing comprehension over "flexibly stratified" formulae)

This is clearly at least as strong as NF, since every theorem of NF is a theorem of Flexible NF.

Is Flexible NF obviously inconsistent, or obviously equiconsistent with NF?

My thoughts:

Could we somehow use flexible stratified comprehension to define the bijection $f: x \mapsto \{x\}$ on e.g. the universal set, and then use that in contradiction with Cantor's Theorem to get inconsistency?