Let x0 of type ι be given.
Let x1 of type ι be given.
Apply ordinal_SNo with
x0.
The subproof is completed by applying H0.
Apply ordinal_SNoLev with
x0.
The subproof is completed by applying H0.
Apply SNoLt_trichotomy_or with
x1,
x0,
SNoLt x1 x0 leaving 4 subgoals.
The subproof is completed by applying H1.
Apply ordinal_SNo with
x0.
The subproof is completed by applying H0.
Assume H5:
or (SNoLt x1 x0) (x1 = x0).
Apply H5 with
SNoLt x1 x0 leaving 2 subgoals.
The subproof is completed by applying H6.
Assume H6: x1 = x0.
Apply FalseE with
SNoLt x1 x0.
Apply In_irref with
x0.
Apply L4 with
λ x2 x3 . x2 ∈ x0.
Apply H6 with
λ x2 x3 . SNoLev x2 ∈ x0.
The subproof is completed by applying H2.
Apply FalseE with
SNoLt x1 x0.
Apply SNoLtE with
x0,
x1,
False leaving 6 subgoals.
The subproof is completed by applying L3.
The subproof is completed by applying H1.
The subproof is completed by applying H5.
Let x2 of type ι be given.
Apply L4 with
λ x2 x3 . x3 ∈ SNoLev x1 ⟶ SNoEq_ x3 x0 x1 ⟶ x3 ∈ x1 ⟶ False.
Apply FalseE with
SNoEq_ x0 x0 x1 ⟶ x0 ∈ x1 ⟶ False.
Apply In_no2cycle with
x0,
SNoLev x1 leaving 2 subgoals.
The subproof is completed by applying H6.
The subproof is completed by applying H2.