Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Assume H2: x2 ∈ x0.
Apply H1 with
λ x3 x4 . x2 ∈ x3.
Apply binunionI1 with
x0,
Sing (Sing 1),
x2.
The subproof is completed by applying H2.
Apply binunionE with
x1,
Sing (Sing 1),
x2,
x2 ∈ x1 leaving 3 subgoals.
The subproof is completed by applying L3.
Assume H4: x2 ∈ x1.
The subproof is completed by applying H4.
Apply FalseE with
x2 ∈ x1.
Apply not_ordinal_Sing1.
Apply SingE with
Sing 1,
x2,
λ x3 x4 . ordinal x3 leaving 2 subgoals.
The subproof is completed by applying H4.
Apply ordinal_Hered with
x0,
x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.