const In : set set prop term iIn = In infix iIn 2000 2000 term nIn = \x:set.\y:set.~ x iIn y const SNo : set prop const SNo_ : set set prop const SNoLev : set set axiom SNoLev_: !x:set.SNo x -> SNo_ (SNoLev x) x const ordinal : set prop const SNoS_ : set set axiom SNoS_I: !x:set.ordinal x -> !y:set.!z:set.z iIn x -> SNo_ z y -> y iIn SNoS_ x const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom SNoLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y < z -> x < z axiom SNoLt_irref: !x:set.~ x < x axiom ordinal_SNoLev: !x:set.ordinal x -> SNoLev x = x axiom In_irref: !x:set.nIn x x axiom FalseE: ~ False 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 SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom SNoLeE: !x:set.!y:set.SNo x -> SNo y -> x <= y -> x < y | x = y var x:set hyp SNo x hyp !y:set.y iIn SNoS_ (SNoLev x) -> y < x hyp ordinal (SNoLev x) hyp SNo (SNoLev x) claim x <= SNoLev x -> SNoLev x = x