const In : set set prop term iIn = In infix iIn 2000 2000 term nIn = \x:set.\y:set.~ x iIn y const nat_p : set prop const ordsucc : set set axiom nat_ordsucc: !x:set.nat_p x -> nat_p (ordsucc x) const ordinal : set prop axiom nat_p_ordinal: !x:set.nat_p x -> ordinal x const omega : set const SNo : set prop const eps_ : set set axiom SNo_eps_: !x:set.x iIn omega -> SNo (eps_ x) const SNoLev : set set axiom SNoLev_eps_: !x:set.x iIn omega -> SNoLev (eps_ x) = ordsucc x axiom In_irref: !x:set.nIn x x axiom FalseE: ~ False axiom SNoLev_ordinal: !x:set.SNo x -> ordinal (SNoLev x) const Empty : set axiom eps_ordinal_In_eq_0: !x:set.!y:set.ordinal y -> y iIn eps_ x -> y = Empty axiom SNoLev_0_eq_0: !x:set.SNo x -> SNoLev x = Empty -> x = Empty axiom In_no2cycle: !x:set.!y:set.x iIn y -> ~ y iIn 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 axiom SNoLt_trichotomy_or_impred: !x:set.!y:set.SNo x -> SNo y -> !P:prop.(x < y -> P) -> (x = y -> P) -> (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.!z:set.Empty < y -> SNo y -> SNo z -> y < z -> SNoLev z iIn eps_ x -> z != Empty lemma !x:set.!y:set.Empty < y -> ordinal (SNoLev y) -> SNo y -> SNoLev y iIn eps_ x -> y != Empty var x:set var y:set hyp nat_p x hyp y iIn SNoS_ (ordsucc x) hyp Empty < y claim x iIn omega -> eps_ x < y