Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H0: x0 ⊆ x1.
Let x2 of type ι be given.
Apply binunionE with
x0,
{(λ x4 . SetAdjoin x4 (Sing 1)) x3|x3 ∈ x0},
x2,
x2 ∈ SNoElts_ x1 leaving 3 subgoals.
The subproof is completed by applying H1.
Assume H2: x2 ∈ x0.
Apply binunionI1 with
x1,
{(λ x4 . SetAdjoin x4 (Sing 1)) x3|x3 ∈ x1},
x2.
Apply H0 with
x2.
The subproof is completed by applying H2.
Apply binunionI2 with
x1,
{(λ x4 . SetAdjoin x4 (Sing 1)) x3|x3 ∈ x1},
x2.
Apply ReplE_impred with
x0,
λ x3 . (λ x4 . SetAdjoin x4 (Sing 1)) x3,
x2,
x2 ∈ {(λ x4 . SetAdjoin x4 (Sing 1)) x3|x3 ∈ x1} leaving 2 subgoals.
The subproof is completed by applying H2.
Let x3 of type ι be given.
Assume H3: x3 ∈ x0.
Apply H4 with
λ x4 x5 . x5 ∈ {(λ x7 . SetAdjoin x7 (Sing 1)) x6|x6 ∈ x1}.
Apply ReplI with
x1,
λ x4 . (λ x5 . SetAdjoin x5 (Sing 1)) x4,
x3.
Apply H0 with
x3.
The subproof is completed by applying H3.