Let x0 of type ι be given.
Let x1 of type ι be given.
Apply SNoEq_I with
x1,
binintersect x0 (SNoElts_ x1),
x0.
Let x2 of type ι be given.
Assume H2: x2 ∈ x1.
Apply iffI with
x2 ∈ binintersect x0 (SNoElts_ x1),
x2 ∈ x0 leaving 2 subgoals.
Apply binintersectE1 with
x0,
SNoElts_ x1,
x2.
The subproof is completed by applying H3.
Assume H3: x2 ∈ x0.
Apply binintersectI with
x0,
SNoElts_ x1,
x2 leaving 2 subgoals.
The subproof is completed by applying H3.
Apply binunionI1 with
x1,
{SetAdjoin x3 (Sing 1)|x3 ∈ x1},
x2.
The subproof is completed by applying H2.