Let x0 of type ι be given.
Let x1 of type ι be given.
Apply and3I with
∀ x2 . x2 ∈ binunion {add_SNo x3 x1|x3 ∈ SNoS_ x0} {add_SNo x0 x3|x3 ∈ SNoS_ x1} ⟶ SNo x2,
∀ x2 . x2 ∈ 0 ⟶ SNo x2,
∀ x2 . x2 ∈ binunion {add_SNo x3 x1|x3 ∈ SNoS_ x0} {add_SNo x0 x3|x3 ∈ SNoS_ x1} ⟶ ∀ x3 . x3 ∈ 0 ⟶ SNoLt x2 x3 leaving 3 subgoals.
Let x2 of type ι be given.
Apply binunionE with
{add_SNo x3 x1|x3 ∈ SNoS_ x0},
{add_SNo x0 x3|x3 ∈ SNoS_ x1},
x2,
SNo x2 leaving 3 subgoals.
The subproof is completed by applying H2.
Apply ReplE_impred with
SNoS_ x0,
λ x3 . add_SNo x3 x1,
x2,
SNo x2 leaving 2 subgoals.
The subproof is completed by applying H3.
Let x3 of type ι be given.
Assume H4:
x3 ∈ SNoS_ x0.
Apply SNoS_E2 with
x0,
x3,
SNo x2 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H4.
Apply H5 with
λ x4 x5 . SNo x5.
Apply SNo_add_SNo with
x3,
x1 leaving 2 subgoals.
The subproof is completed by applying H8.
Apply ordinal_SNo with
x1.
The subproof is completed by applying H1.
Apply ReplE_impred with
SNoS_ x1,
λ x3 . add_SNo x0 x3,
x2,
SNo x2 leaving 2 subgoals.
The subproof is completed by applying H3.
Let x3 of type ι be given.
Assume H4:
x3 ∈ SNoS_ x1.
Apply SNoS_E2 with
x1,
x3,
SNo x2 leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H4.
Apply H5 with
λ x4 x5 . SNo x5.
Apply SNo_add_SNo with
x0,
x3 leaving 2 subgoals.
Apply ordinal_SNo with
x0.
The subproof is completed by applying H0.
The subproof is completed by applying H8.
Let x2 of type ι be given.
Assume H2: x2 ∈ 0.
Apply FalseE with
SNo x2.
Apply EmptyE with
x2.
The subproof is completed by applying H2.
Let x2 of type ι be given.
Let x3 of type ι be given.
Assume H3: x3 ∈ 0.
Apply FalseE with
SNoLt x2 x3.
Apply EmptyE with
x3.
The subproof is completed by applying H3.