Let $T$ be a theory formalized in first order logic with equality and membership and the additional primitive constant symbol $W$, with the following axioms:

1. Extensionality: $\forall z (z \in x \leftrightarrow z \in y) \to x=y$

2. Class comprehension: if $\phi$ is a formula in which $x$ doesn't occur free, then all closures of: $$\exists x \forall y (y \in x \leftrightarrow \phi \land \exists z (y \in z) ) $$; are axioms.

Define: $x=V \iff \forall y (\exists z (y \in z) \to y \in x)$

3. Sub-world: $W \in V$

4. Resemblance: if $\varphi^X$ is a formula in prenex normal form, whose matrix is in $L(=,\in)$, and each variable in its prefix must appear as bounded either $\in X$ or $\subseteq X$, then: $$\varphi^V \leftrightarrow \varphi^W $$, is an axiom.

5. All axioms of "$\sf ZF + \text {CH fails everywhere}"$ relativized to $W$.

What's the consistency strength of $T$?

Let $T$ be a theory formalized in first order logic with equality and membership and the additional primitive constant symbol $W$, with the following axioms:

1. Extensionality: $\forall z (z \in x \leftrightarrow z \in y) \to x=y$

2. Class comprehension: if $\phi$ is a formula in which $x$ doesn't occur free, then all closures of: $$\exists x \forall y (y \in x \leftrightarrow \phi \land \exists z (y \in z) ) $$; are axioms.

   Define: $x=V \iff \forall y (\exists z (y \in z) \to y \in x)$

3. Sub-world: $W \in V$

4. Resemblance: if $\varphi^X$ is a formula in prenex normal form, whose matrix is in $L(=,\in)$, and each variable in its prefix must appear as bounded either $\in X$ or $\subseteq X$, then: $$\varphi^V \leftrightarrow \varphi^W $$, is an axiom.

5. All axioms of "$\sf ZF + \text {CH fails everywhere}"$ relativized to $W$.

What's the consistency strength of $T$?