Let x0 of type ι be given.
Let x1 of type ι be given.
Apply H5 with
add_SNo (ordsucc x0) x1 = ordsucc (add_SNo x0 x1).
Apply dneg with
ordinal x0.
Apply H9.
Apply andI with
TransSet x0,
∀ x2 . x2 ∈ x0 ⟶ TransSet x2 leaving 2 subgoals.
Let x2 of type ι be given.
Assume H10: x2 ∈ x0.
Apply H7 with
x2.
Apply ordsuccI1 with
x0,
x2.
The subproof is completed by applying H10.
Let x3 of type ι be given.
Assume H12: x3 ∈ x2.
Apply L11 with
x3.
The subproof is completed by applying H12.
Apply ordsuccE with
x0,
x3,
x3 ∈ x0 leaving 3 subgoals.
The subproof is completed by applying L13.
Assume H14: x3 ∈ x0.
The subproof is completed by applying H14.
Assume H14: x3 = x0.
Apply FalseE with
x3 ∈ x0.
Apply H9.
Apply H14 with
λ x4 x5 . ordinal x4.
Apply ordinal_Hered with
ordsucc x0,
x3 leaving 2 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying L13.
Let x2 of type ι be given.
Assume H10: x2 ∈ x0.
Apply H8 with
x2.
Apply ordsuccI1 with
x0,
x2.
The subproof is completed by applying H10.
Apply add_SNo_ordinal_SL with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying L9.
The subproof is completed by applying H0.