Let x0 of type ι be given.
Let x1 of type ι be given.
Apply prop_ext_2 with
x0 ⊆ x1,
binunion x0 x1 = x1 leaving 2 subgoals.
Assume H0: x0 ⊆ x1.
Apply set_ext with
binunion x0 x1,
x1 leaving 2 subgoals.
Apply binunion_Subq_min with
x0,
x1,
x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying Subq_ref with x1.
The subproof is completed by applying binunion_Subq_2 with x0, x1.
Apply H0 with
λ x2 x3 . x0 ⊆ x2.
The subproof is completed by applying binunion_Subq_1 with x0, x1.