Let x0 of type ι be given.
Apply omega_nat_p with
x0.
The subproof is completed by applying H0.
Apply and3I with
∀ x1 . x1 ∈ Sing 0 ⟶ SNo x1,
∀ x1 . x1 ∈ {eps_ x2|x2 ∈ x0} ⟶ SNo x1,
∀ x1 . x1 ∈ Sing 0 ⟶ ∀ x2 . x2 ∈ {eps_ x3|x3 ∈ x0} ⟶ SNoLt x1 x2 leaving 3 subgoals.
Let x1 of type ι be given.
Apply SingE with
0,
x1,
λ x2 x3 . SNo x3 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying SNo_0.
Let x1 of type ι be given.
Assume H2:
x1 ∈ {eps_ x2|x2 ∈ x0}.
Apply ReplE_impred with
x0,
eps_,
x1,
SNo x1 leaving 2 subgoals.
The subproof is completed by applying H2.
Let x2 of type ι be given.
Assume H3: x2 ∈ x0.
Apply H4 with
λ x3 x4 . SNo x4.
Apply SNo_eps_ with
x2.
Apply nat_p_omega with
x2.
Apply nat_p_trans with
x0,
x2 leaving 2 subgoals.
The subproof is completed by applying L1.
The subproof is completed by applying H3.
Let x1 of type ι be given.
Let x2 of type ι be given.
Assume H3:
x2 ∈ {eps_ x3|x3 ∈ x0}.
Apply ReplE_impred with
x0,
eps_,
x2,
SNoLt x1 x2 leaving 2 subgoals.
The subproof is completed by applying H3.
Let x3 of type ι be given.
Assume H4: x3 ∈ x0.
Apply SingE with
0,
x1,
λ x4 x5 . SNoLt x5 x2 leaving 2 subgoals.
The subproof is completed by applying H2.
Apply H5 with
λ x4 x5 . SNoLt 0 x5.
Apply SNo_eps_pos with
x3.
Apply nat_p_omega with
x3.
Apply nat_p_trans with
x0,
x3 leaving 2 subgoals.
The subproof is completed by applying L1.
The subproof is completed by applying H4.