Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H2: x0 ⊆ x1.
Apply ordinal_In_Or_Subq with
x0,
x1,
ordsucc x0 ⊆ ordsucc x1 leaving 4 subgoals.
Apply nat_p_ordinal with
x0.
The subproof is completed by applying H0.
Apply nat_p_ordinal with
x1.
The subproof is completed by applying H1.
Assume H3: x0 ∈ x1.
Apply nat_trans with
ordsucc x1,
ordsucc x0 leaving 2 subgoals.
Apply nat_ordsucc with
x1.
The subproof is completed by applying H1.
Apply nat_ordsucc_in_ordsucc with
x1,
x0 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
Assume H3: x1 ⊆ x0.
Claim L4: x0 = x1
Apply set_ext with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Apply L4 with
λ x2 x3 . ordsucc x3 ⊆ ordsucc x1.
The subproof is completed by applying Subq_ref with
ordsucc x1.