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 Empty : set const Power : set set axiom Empty_In_Power: !x:set.Empty iIn Power x const Repl : set (set set) set axiom ReplI: !x:set.!f:set set.!y:set.y iIn x -> f y iIn Repl x f const UPair : set set set const If_i : prop set set set lemma !x:set.!y:set.!z:set.z iIn UPair x y -> If_i (x iIn Power x) x y iIn Repl (Power (Power x)) (\w:set.If_i (x iIn w) x y) -> If_i (x iIn Empty) x y iIn Repl (Power (Power x)) (\w:set.If_i (x iIn w) x y) -> z iIn Repl (Power (Power x)) \w:set.If_i (x iIn w) x y var x:set var y:set var z:set hyp z iIn UPair x y claim If_i (x iIn Power x) x y iIn Repl (Power (Power x)) (\w:set.If_i (x iIn w) x y) -> z iIn Repl (Power (Power x)) \w:set.If_i (x iIn w) x y