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 SNoEq_ = \x:set.\y:set.\z:set.PNoEq_ x (\w:set.w iIn y) \w:set.w iIn z term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y const ordinal : set prop const SNo_ : set set prop term SNo = \x:set.?y:set.ordinal y & SNo_ y x const PNoLt : set (set prop) set (set prop) prop term PNoLt_pwise = \P:set (set prop) prop.\Q:set (set prop) prop.!x:set.ordinal x -> !p:set prop.P x p -> !y:set.ordinal y -> !q:set prop.Q y q -> PNoLt x p y q term PNo_strict_upperbd = \P:set (set prop) prop.\x:set.\p:set prop.!y:set.ordinal y -> !q:set prop.P y q -> PNoLt y q x p term PNo_strict_lowerbd = \P:set (set prop) prop.\x:set.\p:set prop.!y:set.ordinal y -> !q:set prop.P y q -> PNoLt x p y q term PNo_strict_imv = \P:set (set prop) prop.\Q:set (set prop) prop.\x:set.\p:set prop.PNo_strict_upperbd P x p & PNo_strict_lowerbd Q x p term nIn = \x:set.\y:set.~ x iIn y const PNo_rel_strict_imv : (set (set prop) prop) (set (set prop) prop) set (set prop) prop const ordsucc : set set term PNo_rel_strict_split_imv = \P:set (set prop) prop.\Q:set (set prop) prop.\x:set.\p:set prop.PNo_rel_strict_imv P Q (ordsucc x) (\y:set.p y & y != x) & PNo_rel_strict_imv P Q (ordsucc x) \y:set.p y | y = x const SNoLt : set set prop term < = SNoLt infix < 2020 2020 term SNoCutP = \x:set.\y:set.(!z:set.z iIn x -> SNo z) & (!z:set.z iIn y -> SNo z) & !z:set.z iIn x -> !w:set.w iIn y -> z < w term PNo_least_rep = \P:set (set prop) prop.\Q:set (set prop) prop.\x:set.\p:set prop.ordinal x & PNo_strict_imv P Q x p & !y:set.y iIn x -> !q:set prop.~ PNo_strict_imv P Q y q const PSNo : set (set prop) set const PNo_bd : (set (set prop) prop) (set (set prop) prop) set const PNo_pred : (set (set prop) prop) (set (set prop) prop) set prop term SNoCut = \x:set.\y:set.PSNo (PNo_bd (\z:set.\p:set prop.ordinal z & PSNo z p iIn x) \z:set.\p:set prop.ordinal z & PSNo z p iIn y) (PNo_pred (\z:set.\p:set prop.ordinal z & PSNo z p iIn x) \z:set.\p:set prop.ordinal z & PSNo z p iIn 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 PNo_lenbdd = \x:set.\P:set (set prop) prop.!y:set.!p:set prop.P y p -> y iIn x const SNoLev : set set axiom SNo_PSNo_eta: !x:set.SNo x -> x = PSNo (SNoLev x) \y:set.y iIn x lemma !x:set.!y:set.!z:set.ordinal (PNo_bd (\w:set.\p:set prop.ordinal w & PSNo w p iIn x) \w:set.\p:set prop.ordinal w & PSNo w p iIn y) -> PNo_strict_lowerbd (\w:set.\p:set prop.ordinal w & PSNo w p iIn y) (PNo_bd (\w:set.\p:set prop.ordinal w & PSNo w p iIn x) \w:set.\p:set prop.ordinal w & PSNo w p iIn y) (PNo_pred (\w:set.\p:set prop.ordinal w & PSNo w p iIn x) \w:set.\p:set prop.ordinal w & PSNo w p iIn y) -> z iIn y -> SNo z -> ordinal (SNoLev z) -> z = PSNo (SNoLev z) (\w:set.w iIn z) -> PSNo (PNo_bd (\w:set.\p:set prop.ordinal w & PSNo w p iIn x) \w:set.\p:set prop.ordinal w & PSNo w p iIn y) (PNo_pred (\w:set.\p:set prop.ordinal w & PSNo w p iIn x) \w:set.\p:set prop.ordinal w & PSNo w p iIn y) < z var x:set var y:set var z:set hyp ordinal (PNo_bd (\w:set.\p:set prop.ordinal w & PSNo w p iIn x) \w:set.\p:set prop.ordinal w & PSNo w p iIn y) hyp PNo_strict_lowerbd (\w:set.\p:set prop.ordinal w & PSNo w p iIn y) (PNo_bd (\w:set.\p:set prop.ordinal w & PSNo w p iIn x) \w:set.\p:set prop.ordinal w & PSNo w p iIn y) (PNo_pred (\w:set.\p:set prop.ordinal w & PSNo w p iIn x) \w:set.\p:set prop.ordinal w & PSNo w p iIn y) hyp z iIn y hyp SNo z claim ordinal (SNoLev z) -> PSNo (PNo_bd (\w:set.\p:set prop.ordinal w & PSNo w p iIn x) \w:set.\p:set prop.ordinal w & PSNo w p iIn y) (PNo_pred (\w:set.\p:set prop.ordinal w & PSNo w p iIn x) \w:set.\p:set prop.ordinal w & PSNo w p iIn y) < z