Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H1: x1 ∈ x0.
Apply ordinal_ordsucc with
x1.
Apply ordinal_Hered with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply ordinal_trichotomy_or with
x0,
ordsucc x1,
or (ordsucc x1 ∈ x0) (x0 = ordsucc x1) leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L2.
Apply H3 with
or (ordsucc x1 ∈ x0) (x0 = ordsucc x1) leaving 2 subgoals.
Apply FalseE with
or (ordsucc x1 ∈ x0) (x0 = ordsucc x1).
Apply ordsuccE with
x1,
x0,
False leaving 3 subgoals.
The subproof is completed by applying H4.
Assume H5: x0 ∈ x1.
Apply In_no2cycle with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H1.
Assume H5: x0 = x1.
Apply In_irref with
x0.
Apply H5 with
λ x2 x3 . x3 ∈ x0.
The subproof is completed by applying H1.
Apply orIR with
ordsucc x1 ∈ x0,
x0 = ordsucc x1.
The subproof is completed by applying H4.
Apply orIL with
ordsucc x1 ∈ x0,
x0 = ordsucc x1.
The subproof is completed by applying H3.