Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H2: x0 ⊆ x1.
Let x2 of type ι be given.
Apply ordinal_trichotomy_or_impred with
x0,
x1,
add_nat x0 x2 ⊆ add_nat x1 x2 leaving 5 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 H4: x0 ∈ x1.
Apply unknownprop_eeaa5555ccfaf9be2474522165cc658a4c21b3dcbef964c1d1aad1f792298727 with
x1,
x0,
x2 leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H4.
The subproof is completed by applying H3.
Apply nat_trans with
add_nat x1 x2,
add_nat x0 x2 leaving 2 subgoals.
Apply add_nat_p with
x1,
x2 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
The subproof is completed by applying L5.
Assume H4: x0 = x1.
Apply H4 with
λ x3 x4 . add_nat x4 x2 ⊆ add_nat x1 x2.
The subproof is completed by applying Subq_ref with
add_nat x1 x2.
Assume H4: x1 ∈ x0.
Apply FalseE with
add_nat x0 x2 ⊆ add_nat x1 x2.
Apply In_irref with
x1.
Apply H2 with
x1.
The subproof is completed by applying H4.