Аксиома бесконечности

${\displaystyle \exists a_{\infty }\ (\exists a_{\varnothing }\ (a_{\varnothing }\in a_{\infty }\ \land \ \forall b\ (b\notin a_{\varnothing }))\ \ \land \ \ \forall b\exists c\forall d\ (b\in a_{\infty }\to (c\in a_{\infty }\ \land \ (d\in c\leftrightarrow d\in b\ \lor \ d=b))))}$

${\displaystyle \exists a_{\infty }\ (\exists a_{\varnothing }\ (a_{\varnothing }\in a_{\infty }\ \land \ \forall b\ (b\notin a_{\varnothing }))\ \ \land \ \ \forall b\forall c\exists d\ (b\in a_{\infty }\to ((d\in c\leftrightarrow d\in b\ \lor \ d=b)\to c\in a_{\infty })))}$