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 axiom SNoLe_ref: !x:set.x <= x const ordsucc : set set const Empty : set axiom cases_1: !x:set.x iIn ordsucc Empty -> !p:set prop.p Empty -> p x var x:set var f:set set hyp !y:set.y iIn x -> SNo y hyp !y:set.y iIn ordsucc Empty -> f y iIn x hyp !y:set.y iIn x -> ?z:set.z iIn ordsucc Empty & f z = y claim f Empty iIn x -> f Empty iIn x & SNo (f Empty) & !y:set.y iIn x -> SNo y -> y <= f Empty