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 ordinal : set prop const omega : set axiom omega_ordinal: ordinal omega const finite : set prop const SNoS_ : set set axiom SNoS_finite: !x:set.x iIn omega -> finite (SNoS_ x) const SNo : set prop const SNoLev : set set axiom SNoS_I2: !x:set.!y:set.SNo x -> SNo y -> SNoLev x iIn SNoLev y -> x iIn SNoS_ (SNoLev y) const SNoR : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom SNoR_E: !x:set.SNo x -> !y:set.y iIn SNoR x -> !P:prop.(SNo y -> SNoLev y iIn SNoLev x -> x < y -> P) -> P axiom Subq_finite: !x:set.finite x -> !y:set.Subq y x -> finite y const SNo_ : set set prop axiom SNoS_E2: !x:set.ordinal x -> !y:set.y iIn SNoS_ x -> !P:prop.(SNoLev y iIn x -> ordinal (SNoLev y) -> SNo y -> SNo_ (SNoLev y) y -> P) -> P claim !x:set.x iIn SNoS_ omega -> finite (SNoR x)