const In : set set prop term iIn = In infix iIn 2000 2000 term PNoEq_ = \x:set.\p:set prop.\q:set prop.!y:set.y iIn x -> (p y <-> q y) term PNoLt_ = \x:set.\p:set prop.\q:set prop.?y:set.y iIn x & (PNoEq_ y p q & ~ p y & q y) term nIn = \x:set.\y:set.~ x iIn y axiom not_all_ex_demorgan_i: !p:set prop.~ (!x:set.p x) -> ?x:set.~ p x axiom xm: !P:prop.P | ~ P const ordinal : set prop axiom ordinal_ind: !p:set prop.(!x:set.ordinal x -> (!y:set.y iIn x -> p y) -> p x) -> !x:set.ordinal x -> p x lemma !p:set prop.!q:set prop.!x:set.ordinal x -> (!y:set.y iIn x -> PNoLt_ y p q | PNoEq_ y p q | PNoLt_ y q p) -> ~ PNoEq_ x p q -> (?y:set.~ (y iIn x -> (p y <-> q y))) -> PNoLt_ x p q | PNoEq_ x p q | PNoLt_ x q p claim !p:set prop.!q:set prop.!x:set.ordinal x -> PNoLt_ x p q | PNoEq_ x p q | PNoLt_ x q p