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 SNo : set prop axiom ordinal_SNo: !x:set.ordinal x -> SNo x axiom In_irref: !x:set.nIn x x axiom FalseE: ~ False const binintersect : set set set axiom binintersectE1: !x:set.!y:set.!z:set.z iIn binintersect x y -> z iIn x axiom In_no2cycle: !x:set.!y:set.x iIn y -> ~ y iIn x const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const SNoLev : 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: !x:set.!y:set.SNo x -> SNo y -> x < y | x = y | y < x var x:set var y:set hyp ordinal x hyp SNo y hyp SNoLev y iIn x hyp SNo x claim SNoLev x = x -> y < x