Is

$$U_{\omega}=\Big\{x\mid\forall z\Big(\big(\emptyset\in z\wedge \forall u, v\;(u,v\in z\rightarrow\{w\mid w\in u\vee w=v\}\in z)\big)\rightarrow x\in z\Big)\Big\}$$

identical with the set $V_{\omega}$ of hereditarily finite sets, i.e. the level $\omega$ of the cumulative hierarchy?

Is

$$\mathrm{U}_{\omega}=\Big\{x\mid\forall z\Big(\big(\emptyset\in z\wedge \forall u, v\;(u,v\in z\rightarrow\{w\mid w\in u\vee w=v\}\in z)\big)\rightarrow x\in z\Big)\Big\}$$

identical with the set $\mathrm{V}_{\omega}$ of hereditarily finite sets, i.e. the level $\omega$ of the cumulative hierarchy?

Is

$$U_{\omega}=\Big\{x\mid\forall z\Big(\big(\emptyset\in z\wedge \forall u, v\;(u,v\in z\rightarrow\{w\mid w\in u\vee w=u\}\in z)\big)\rightarrow x\in z\Big)\Big\}$$

identical with the set $V_{\omega}$ of hereditarily finite sets, i.e. the level $\omega$ of the cumulative hierarchy?

Is

$$U_{\omega}=\Big\{x\mid\forall z\Big(\big(\emptyset\in z\wedge \forall u, v\;(u,v\in z\rightarrow\{w\mid w\in u\vee w=v\}\in z)\big)\rightarrow x\in z\Big)\Big\}$$

identical with the set $V_{\omega}$ of hereditarily finite sets, i.e. the level $\omega$ of the cumulative hierarchy?

Is

$U_{\omega}=\{x|\forall z((\emptyset\in z\wedge \forall u, v(u,v\in z\rightarrow\{w|w\in u\vee w=u\}\in z))\rightarrow x\in z)\}$

identical with the set $V_{\omega}$ of hereditarily finite sets, i.e. the level $\omega$ of the cumulative hierarchy?

Is

$$U_{\omega}=\Big\{x\mid\forall z\Big(\big(\emptyset\in z\wedge \forall u, v\;(u,v\in z\rightarrow\{w\mid w\in u\vee w=u\}\in z)\big)\rightarrow x\in z\Big)\Big\}$$

identical with the set $V_{\omega}$ of hereditarily finite sets, i.e. the level $\omega$ of the cumulative hierarchy?