Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H2: x0 ⊆ x1.
Apply ordinal_In_Or_Subq with
ordsucc x1,
ordsucc x0,
ordsucc x0 ⊆ ordsucc x1 leaving 4 subgoals.
Apply ordinal_ordsucc with
x1.
The subproof is completed by applying H1.
Apply ordinal_ordsucc with
x0.
The subproof is completed by applying H0.
Apply FalseE with
ordsucc x0 ⊆ ordsucc x1.
Apply ordsuccE with
x0,
ordsucc x1,
False leaving 3 subgoals.
The subproof is completed by applying H3.
Apply In_no2cycle with
x1,
ordsucc x1 leaving 2 subgoals.
The subproof is completed by applying ordsuccI2 with x1.
Apply H2 with
ordsucc x1.
The subproof is completed by applying H4.
Apply In_irref with
x1.
Apply H2 with
x1.
Apply H4 with
λ x2 x3 . x1 ∈ x2.
The subproof is completed by applying ordsuccI2 with x1.
The subproof is completed by applying H3.