const In : set set prop term iIn = In infix iIn 2000 2000 term nIn = \x:set.\y:set.~ x iIn y const Empty : set axiom EmptyE: !x:set.nIn x Empty axiom FalseE: ~ False const Subq : set set prop const ordsucc : set set axiom ordsuccI1: !x:set.Subq x (ordsucc x) axiom Subq_tra: !x:set.!y:set.!z:set.Subq x y -> Subq y z -> Subq x z axiom ordsuccE: !x:set.!y:set.y iIn ordsucc x -> y iIn x | y = x const nat_p : set prop axiom nat_ind: !p:set prop.p Empty -> (!x:set.nat_p x -> p x -> p (ordsucc x)) -> !x:set.nat_p x -> p x claim !x:set.nat_p x -> !y:set.y iIn x -> Subq y x