Let x0 of type ι be given.
Let x1 of type ι be given.
Apply or3E with
x0 ∈ x1,
x0 = x1,
x1 ∈ x0,
or (x0 ∈ x1) (x1 ⊆ x0) leaving 4 subgoals.
Apply ordinal_trichotomy_or with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Assume H2: x0 ∈ x1.
Apply orIL with
x0 ∈ x1,
x1 ⊆ x0.
The subproof is completed by applying H2.
Assume H2: x0 = x1.
Apply orIR with
x0 ∈ x1,
x1 ⊆ x0.
Apply H2 with
λ x2 x3 . x1 ⊆ x3.
The subproof is completed by applying Subq_ref with x1.
Assume H2: x1 ∈ x0.
Apply orIR with
x0 ∈ x1,
x1 ⊆ x0.
Apply ordinal_TransSet with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.