const In : set set prop term iIn = In infix iIn 2000 2000 const SNo : set prop const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 term SNo_max_of = \x:set.\y:set.y iIn x & SNo y & !z:set.z iIn x -> SNo z -> z <= y const bij : set set (set set) prop term equip = \x:set.\y:set.?f:set set.bij x y f const omega : set term finite = \x:set.?y:set.y iIn omega & equip x y term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y term nIn = \x:set.\y:set.~ x iIn y const nat_p : set prop axiom omega_nat_p: !x:set.x iIn omega -> nat_p x const Empty : set axiom EmptyE: !x:set.nIn x Empty axiom Empty_eq: !x:set.(!y:set.nIn y x) -> x = Empty axiom bijE: !x:set.!y:set.!f:set set.bij x y f -> !P:prop.((!z:set.z iIn x -> f z iIn y) -> (!z:set.z iIn x -> !w:set.w iIn x -> f z = f w -> z = w) -> (!z:set.z iIn y -> ?w:set.w iIn x & f w = z) -> P) -> P axiom FalseE: ~ False const ordsucc : set set axiom nat_inv: !x:set.nat_p x -> x = Empty | ?y:set.nat_p y & x = ordsucc y claim (!x:set.nat_p x -> !y:set.(!z:set.z iIn y -> SNo z) -> equip y (ordsucc x) -> ?z:set.SNo_max_of y z) -> !x:set.(!y:set.y iIn x -> SNo y) -> finite x -> x != Empty -> ?y:set.SNo_max_of x y