const In : set set prop term iIn = In infix iIn 2000 2000 term nIn = \x:set.\y:set.~ x iIn y term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y const If_i : prop set set set axiom If_i_1: !P:prop.!x:set.!y:set.P -> If_i P x y = x axiom If_i_0: !P:prop.!x:set.!y:set.~ P -> If_i P x y = y axiom xm: !P:prop.P | ~ P const ZF_closed : set prop const Power : set set const Repl : set (set set) set lemma !x:set.!y:set.!z:set.ZF_closed x -> y iIn x -> z iIn x -> Power (Power y) iIn x -> (!w:set.w iIn Power (Power y) -> If_i (y iIn w) y z iIn x) -> Repl (Power (Power y)) (\w:set.If_i (y iIn w) y z) iIn x var x:set var y:set var z:set hyp ZF_closed x hyp y iIn x hyp z iIn x claim Power (Power y) iIn x -> Repl (Power (Power y)) (\w:set.If_i (y iIn w) y z) iIn x