Let $x^*$ be the set of all cardinalities that are strictly larger than the cardinality of $x$ and strictly smaller than the cardinality of $P(x)$. Formally, that is:
Define: $s=x^* \iff \forall y (y \in s \leftrightarrow \exists k (|x| < |k| < |P(x)| \wedge y=|k|))$
Are the following rules consistent with ZFC? if so, what's their consistency strength?
$\forall x (|x|>2 \to |x| < |x^*| <|P(x)| )$
$\forall x (|x|>2 \to ||x|^{+*}| > |P(x)|)$
$\forall x \forall y (|x| > |y| \wedge |y|>2 \to |x^*| > |P(y)|)$
Where $|x|^+$ is the successor cardinal of the cardinality of $x$.
Where cardinality $``||"$ is defined in the customary manner after Von Neumann's.
Also for better reading the above rules can be written in terms of cardinals as:
$\forall \kappa \, (\kappa >2 \to \kappa < |\kappa^*| < 2^\kappa )$
$\forall \kappa \,(\kappa >2 \to |(\kappa^+)^*| > 2^\kappa)$
$\forall \kappa \,\forall \lambda\, (\kappa > \lambda >2 \to |\kappa^*| > 2^\lambda)$
Where $``<",``>"$ are the known cardinal inequalities; and $\kappa^*=\{\lambda\mid \kappa < \lambda < 2^\kappa \}$