const In : set set prop term iIn = In infix iIn 2000 2000 term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y const nat_p : set prop const ordsucc : set set axiom nat_ordsucc_trans: !x:set.nat_p x -> !y:set.y iIn ordsucc x -> Subq y x const Union : set set axiom UnionE_impred: !x:set.!y:set.y iIn Union x -> !P:prop.(!z:set.y iIn z -> z iIn x -> P) -> P axiom ordsuccI2: !x:set.x iIn ordsucc x axiom UnionI: !x:set.!y:set.!z:set.y iIn z -> z iIn x -> y iIn Union x axiom set_ext: !x:set.!y:set.Subq x y -> Subq y x -> x = y axiom nat_complete_ind: !p:set prop.(!x:set.nat_p x -> (!y:set.y iIn x -> p y) -> p x) -> !x:set.nat_p x -> p x claim !x:set.nat_p x -> Union (ordsucc x) = x