const In : set set prop term iIn = In infix iIn 2000 2000 term nIn = \x:set.\y:set.~ x iIn y const ordinal : set prop const ordsucc : set set const omega : set axiom ordsucc_omega_ordinal: ordinal (ordsucc omega) const SNo : set prop axiom SNo_omega: SNo omega const SNoLev : set set axiom SNoLev_ordinal: !x:set.SNo x -> ordinal (SNoLev x) axiom omega_ordinal: ordinal omega axiom In_no2cycle: !x:set.!y:set.x iIn y -> ~ y iIn x axiom In_irref: !x:set.nIn x x axiom ordsuccE: !x:set.!y:set.y iIn ordsucc x -> y iIn x | y = x axiom FalseE: ~ False axiom ordinal_SNoLev: !x:set.ordinal x -> SNoLev x = x const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const binintersect : set set set const SNoEq_ : set set set prop axiom SNoLtE: !x:set.!y:set.SNo x -> SNo y -> x < y -> !P:prop.(!z:set.SNo z -> SNoLev z iIn binintersect (SNoLev x) (SNoLev y) -> SNoEq_ (SNoLev z) z x -> SNoEq_ (SNoLev z) z y -> x < z -> z < y -> nIn (SNoLev z) x -> SNoLev z iIn y -> P) -> (SNoLev x iIn SNoLev y -> SNoEq_ (SNoLev x) x y -> SNoLev x iIn y -> P) -> (SNoLev y iIn SNoLev x -> SNoEq_ (SNoLev y) x y -> nIn (SNoLev y) x -> P) -> P const SNoS_ : set set 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 lemma !x:set.!y:set.SNo x -> SNo y -> x < y -> SNoLev y iIn omega -> ordinal (SNoLev y) -> ?z:set.z iIn omega & x < z lemma !x:set.SNo x -> SNoLev x iIn omega -> ordinal (SNoLev x) -> ?y:set.y iIn omega & x < y claim !x:set.x iIn SNoS_ (ordsucc omega) -> x < omega -> ?y:set.y iIn omega & x < y