Let x0 of type ι be given.
Let x1 of type ι be given.
Apply or3E with
x0 ∈ x1,
x0 = x1,
x1 ∈ x0,
exactly1of3 (x0 ∈ x1) (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 exactly1of3_I1 with
x0 ∈ x1,
x0 = x1,
x1 ∈ x0 leaving 3 subgoals.
The subproof is completed by applying H2.
Assume H3: x0 = x1.
Apply In_irref with
x0.
Apply H3 with
λ x2 x3 . x0 ∈ x3.
The subproof is completed by applying H2.
Assume H3: x1 ∈ x0.
Apply In_no2cycle with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Assume H2: x0 = x1.
Apply exactly1of3_I2 with
x0 ∈ x1,
x0 = x1,
x1 ∈ x0 leaving 3 subgoals.
Apply H2 with
λ x2 x3 . nIn x3 x1.
The subproof is completed by applying In_irref with x1.
The subproof is completed by applying H2.
Apply H2 with
λ x2 x3 . nIn x1 x3.
The subproof is completed by applying In_irref with x1.
Assume H2: x1 ∈ x0.
Apply exactly1of3_I3 with
x0 ∈ x1,
x0 = x1,
x1 ∈ x0 leaving 3 subgoals.
Assume H3: x0 ∈ x1.
Apply In_no2cycle with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H2.
Assume H3: x0 = x1.
Apply In_irref with
x0.
Apply H3 with
λ x2 x3 . x3 ∈ x0.
The subproof is completed by applying H2.
The subproof is completed by applying H2.