Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H0: x1 ∈ x0.
Apply set_ext with
x0,
binunion (setminus x0 (Sing x1)) (Sing x1) leaving 2 subgoals.
Let x2 of type ι be given.
Assume H1: x2 ∈ x0.
Apply xm with
x2 = x1,
x2 ∈ binunion (setminus x0 (Sing x1)) (Sing x1) leaving 2 subgoals.
Assume H2: x2 = x1.
Apply binunionI2 with
setminus x0 (Sing x1),
Sing x1,
x2.
Apply H2 with
λ x3 x4 . x4 ∈ Sing x1.
The subproof is completed by applying SingI with x1.
Assume H2: x2 = x1 ⟶ ∀ x3 : ο . x3.
Apply binunionI1 with
setminus x0 (Sing x1),
Sing x1,
x2.
Apply setminusI with
x0,
Sing x1,
x2 leaving 2 subgoals.
The subproof is completed by applying H1.
Apply H2.
Apply SingE with
x1,
x2.
The subproof is completed by applying H3.
Let x2 of type ι be given.
Apply binunionE with
setminus x0 (Sing x1),
Sing x1,
x2,
x2 ∈ x0 leaving 3 subgoals.
The subproof is completed by applying H1.
Apply setminusE1 with
x0,
Sing x1,
x2.
The subproof is completed by applying H2.
Apply SingE with
x1,
x2,
λ x3 x4 . x4 ∈ x0 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H0.