Let x0 of type ι be given.
Let x1 of type ι be given.
Apply nat_inv with
x1,
and (x0 = 0) (x1 = 0) leaving 3 subgoals.
Apply omega_nat_p with
x1.
The subproof is completed by applying H1.
Assume H3: x1 = 0.
Apply andI with
x0 = 0,
x1 = 0 leaving 2 subgoals.
Apply H2 with
λ x2 x3 . x3 = 0.
Apply H3 with
λ x2 x3 . minus_SNo x3 = 0.
The subproof is completed by applying minus_SNo_0.
The subproof is completed by applying H3.
Apply FalseE with
and (x0 = 0) (x1 = 0).
Apply H3 with
False.
Let x2 of type ι be given.
Apply H4 with
False.
Apply SNoLt_irref with
0.
Apply SNoLeLt_tra with
0,
x0,
0 leaving 5 subgoals.
The subproof is completed by applying SNo_0.
Apply omega_SNo with
x0.
The subproof is completed by applying H0.
The subproof is completed by applying SNo_0.
Apply ordinal_Subq_SNoLe with
0,
x0 leaving 3 subgoals.
The subproof is completed by applying ordinal_Empty.
Apply nat_p_ordinal with
x0.
Apply omega_nat_p with
x0.
The subproof is completed by applying H0.
The subproof is completed by applying Subq_Empty with x0.
Apply H2 with
λ x3 x4 . SNoLt x4 0.
Apply minus_SNo_Lt_contra1 with
0,
x1 leaving 3 subgoals.
The subproof is completed by applying SNo_0.
Apply omega_SNo with
x1.
The subproof is completed by applying H1.
Apply minus_SNo_0 with
λ x3 x4 . SNoLt x4 x1.
Apply ordinal_In_SNoLt with
x1,
0 leaving 2 subgoals.
Apply nat_p_ordinal with
x1.
Apply omega_nat_p with
x1.
The subproof is completed by applying H1.
Apply H6 with
λ x3 x4 . 0 ∈ x4.
Apply nat_0_in_ordsucc with
x2.
The subproof is completed by applying H5.