const SNo_rec_ii : (set (set set set) set set) set set set const If_i : prop set set set const SNo : set prop const SNo_rec_i : (set (set set) set) set set const Empty : set term SNo_rec2 = \P:set set (set set set) set.SNo_rec_ii \x:set.\g:set set set.\y:set.If_i (SNo y) (SNo_rec_i (\z:set.\f:set set.P x z \w:set.\u:set.If_i (w = x) (f u) (g w u)) y) Empty const In : set set prop term iIn = In infix iIn 2000 2000 const SNoS_ : set set const SNoLev : set set axiom SNo_rec2_G_prop: !P:set set (set set set) set.(!x:set.SNo x -> !y:set.SNo y -> !g:set set set.!h:set set set.(!z:set.z iIn SNoS_ (SNoLev x) -> !w:set.SNo w -> g z w = h z w) -> (!z:set.z iIn SNoS_ (SNoLev y) -> g x z = h x z) -> P x y g = P x y h) -> !x:set.SNo x -> !g:set set set.!h:set set set.(!y:set.y iIn SNoS_ (SNoLev x) -> g y = h y) -> !y:set.SNo y -> !f:set set.!f2:set set.(!z:set.z iIn SNoS_ (SNoLev y) -> f z = f2 z) -> P x y (\z:set.\w:set.If_i (z = x) (f w) (g z w)) = P x y \z:set.\w:set.If_i (z = x) (f2 w) (h z w) axiom SNo_rec2_eq_1: !P:set set (set set set) set.(!x:set.SNo x -> !y:set.SNo y -> !g:set set set.!h:set set set.(!z:set.z iIn SNoS_ (SNoLev x) -> !w:set.SNo w -> g z w = h z w) -> (!z:set.z iIn SNoS_ (SNoLev y) -> g x z = h x z) -> P x y g = P x y h) -> !x:set.SNo x -> !g:set set set.!y:set.SNo y -> SNo_rec_i (\z:set.\f:set set.P x z \w:set.\u:set.If_i (w = x) (f u) (g w u)) y = P x y \z:set.\w:set.If_i (z = x) (SNo_rec_i (\u:set.\f:set set.P x u \v:set.\x2:set.If_i (v = x) (f x2) (g v x2)) w) (g z w) const ordinal : set prop const SNo_ : set set prop axiom SNoS_E2: !x:set.ordinal x -> !y:set.y iIn SNoS_ x -> !P:prop.(SNoLev y iIn x -> ordinal (SNoLev y) -> SNo y -> SNo_ (SNoLev y) y -> P) -> P 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 axiom If_i_1: !P:prop.!x:set.!y:set.P -> If_i P x y = x axiom If_i_0: !P:prop.!x:set.!y:set.~ P -> If_i P x y = y axiom xm: !P:prop.P | ~ P axiom func_ext: !f:set set.!f2:set set.(!x:set.f x = f2 x) -> f = f2 lemma !P:set set (set set set) set.!x:set.!y:set.(!z:set.SNo z -> !w:set.SNo w -> !g:set set set.!h:set set set.(!u:set.u iIn SNoS_ (SNoLev z) -> !v:set.SNo v -> g u v = h u v) -> (!u:set.u iIn SNoS_ (SNoLev w) -> g z u = h z u) -> P z w g = P z w h) -> SNo x -> SNo y -> (!z:set.SNo z -> !g:set set set.!h:set set set.(!w:set.w iIn SNoS_ (SNoLev z) -> g w = h w) -> (\w:set.If_i (SNo w) (SNo_rec_i (\u:set.\f:set set.P z u \v:set.\x2:set.If_i (v = z) (f x2) (g v x2)) w) Empty) = \w:set.If_i (SNo w) (SNo_rec_i (\u:set.\f:set set.P z u \v:set.\x2:set.If_i (v = z) (f x2) (h v x2)) w) Empty) -> SNo_rec_ii (\z:set.\g:set set set.\w:set.If_i (SNo w) (SNo_rec_i (\u:set.\f:set set.P z u \v:set.\x2:set.If_i (v = z) (f x2) (g v x2)) w) Empty) x y = P x y (SNo_rec2 P) lemma !P:set set (set set set) set.!x:set.!g:set set set.!h:set set set.!y:set.(!z:set.SNo z -> !w:set.SNo w -> !g2:set set set.!h2:set set set.(!u:set.u iIn SNoS_ (SNoLev z) -> !v:set.SNo v -> g2 u v = h2 u v) -> (!u:set.u iIn SNoS_ (SNoLev w) -> g2 z u = h2 z u) -> P z w g2 = P z w h2) -> SNo x -> (!z:set.z iIn SNoS_ (SNoLev x) -> g z = h z) -> SNo y -> (!z:set.ordinal z -> !w:set.w iIn SNoS_ z -> SNo_rec_i (\u:set.\f:set set.P x u \v:set.\x2:set.If_i (v = x) (f x2) (g v x2)) w = SNo_rec_i (\u:set.\f:set set.P x u \v:set.\x2:set.If_i (v = x) (f x2) (h v x2)) w) -> SNo_rec_i (\z:set.\f:set set.P x z \w:set.\u:set.If_i (w = x) (f u) (g w u)) y = SNo_rec_i (\z:set.\f:set set.P x z \w:set.\u:set.If_i (w = x) (f u) (h w u)) y claim !P:set set (set set set) set.(!x:set.SNo x -> !y:set.SNo y -> !g:set set set.!h:set set set.(!z:set.z iIn SNoS_ (SNoLev x) -> !w:set.SNo w -> g z w = h z w) -> (!z:set.z iIn SNoS_ (SNoLev y) -> g x z = h x z) -> P x y g = P x y h) -> !x:set.SNo x -> !y:set.SNo y -> SNo_rec2 P x y = P x y (SNo_rec2 P)