Let x0 of type ι be given.
Let x1 of type ι be given.
Apply binunionE with
Sing 0,
{(λ x3 . SetAdjoin x3 (Sing 1)) (ordsucc x2)|x2 ∈ x0},
x1,
x1 = 0 leaving 3 subgoals.
The subproof is completed by applying H1.
Apply SingE with
0,
x1.
The subproof is completed by applying H2.
Apply FalseE with
x1 = 0.
Apply ReplE_impred with
x0,
λ x2 . (λ x3 . SetAdjoin x3 (Sing 1)) (ordsucc x2),
x1,
False leaving 2 subgoals.
The subproof is completed by applying H2.
Let x2 of type ι be given.
Assume H3: x2 ∈ x0.
Apply tagged_not_ordinal with
ordsucc x2.
Apply H4 with
λ x3 x4 . ordinal x3.
The subproof is completed by applying H0.