Let $SF$ be the schema of stratified comprehension.

Take the theory $SF + Infinity + Choice + \text {Extensionality fails everywhere}$.

Are the following consistent with this theory?

$\forall X (|X| \leq |P_1(X)|)$

$\forall X (Infinite(X) \to |X|=|P_1(X)|)$

It is known that these two statements fail in NFU, since NFU proves $|P_1(X)| < |X|$ for some sets, which makes it guilty of committing cardinality error of the first kind.

However the known proof of that error in NFU is Extensionality dependent, hence the question. It might be the case that removal of Extensionality may result in avoidance of first kind cardinality error.

It is known that this theory can interpret $NFU + Infinity + Choice$ because SF can interpret NFU a proof due to Marcel Crabbé. An equivalent proof of NFU being interpretable form SF is present here 15-17.