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 term nIn = \x:set.\y:set.~ x iIn y term SNo_min_of = \x:set.\y:set.y iIn x & SNo y & !z:set.z iIn x -> SNo z -> y <= z term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y term TransSet = \x:set.!y:set.y iIn x -> Subq y x const nat_p : set prop const omega : set axiom nat_p_omega: !x:set.nat_p x -> x iIn omega const diadic_rational_p : set prop axiom omega_diadic_rational_p: !x:set.x iIn omega -> diadic_rational_p x axiom FalseE: ~ False const ordinal : set prop const SNoLev : set set const SNoR : set set var x:set var y:set hyp nat_p x hyp SNoLev y = x hyp ~ diadic_rational_p y hyp ordinal y claim y = x -> ?z:set.SNo_min_of (SNoR y) z