const In : set set prop term iIn = In infix iIn 2000 2000 const SNo : set prop 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 Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y term nIn = \x:set.\y:set.~ x iIn y const nat_p : set prop axiom nat_p_trans: !x:set.nat_p x -> !y:set.y iIn x -> nat_p y const omega : set axiom nat_p_omega: !x:set.nat_p x -> x iIn omega const ordinal : set prop axiom nat_p_ordinal: !x:set.nat_p x -> ordinal x const ordsucc : set set axiom ordinal_ordsucc_In: !x:set.ordinal x -> !y:set.y iIn x -> ordsucc y iIn ordsucc x const SNoLev : set set const eps_ : set set axiom SNoLev_eps_: !x:set.x iIn omega -> SNoLev (eps_ x) = ordsucc x const Repl : set (set set) set axiom ReplE_impred: !x:set.!f:set set.!y:set.y iIn Repl x f -> !P:prop.(!z:set.z iIn x -> y = f z -> P) -> P const famunion : set (set set) set axiom famunionE_impred: !x:set.!f:set set.!y:set.y iIn famunion x f -> !P:prop.(!z:set.z iIn x -> y iIn f z -> P) -> P axiom FalseE: ~ False axiom ordsuccI2: !x:set.x iIn ordsucc x axiom ReplI: !x:set.!f:set set.!y:set.y iIn x -> f y iIn Repl x f axiom famunionI: !x:set.!f:set set.!y:set.!z:set.y iIn x -> z iIn f y -> z iIn famunion x f const Empty : set axiom nat_inv: !x:set.nat_p x -> x = Empty | ?y:set.nat_p y & x = ordsucc y axiom set_ext: !x:set.!y:set.Subq x y -> Subq y x -> x = y const Sing : set set const SNoCut : set set set lemma !x:set.x iIn omega -> nat_p x -> SNoCutP (Sing Empty) (Repl x eps_) -> (!y:set.y iIn Repl x eps_ -> SNo y) -> famunion (Sing Empty) (\y:set.ordsucc (SNoLev y)) = ordsucc Empty -> (x != Empty -> famunion (Repl x eps_) (\y:set.ordsucc (SNoLev y)) = ordsucc x) -> eps_ x = SNoCut (Sing Empty) (Repl x eps_) lemma !x:set.!y:set.!z:set.nat_p x -> z iIn x -> y iIn ordsucc (ordsucc z) -> ordsucc z iIn ordsucc x -> y iIn ordsucc x var x:set hyp x iIn omega hyp nat_p x hyp SNoCutP (Sing Empty) (Repl x eps_) hyp !y:set.y iIn Repl x eps_ -> SNo y claim famunion (Sing Empty) (\y:set.ordsucc (SNoLev y)) = ordsucc Empty -> eps_ x = SNoCut (Sing Empty) (Repl x eps_)