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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply set_ext with setminus x0 (setminus x1 x2), binunion (setminus x0 x1) (binintersect x0 x2) leaving 2 subgoals.
Let x3 of type ι be given.
Assume H0: x3 ∈ setminus x0 (setminus x1 x2).
Apply setminusE with x0, setminus x1 x2, x3, x3 ∈ binunion (setminus x0 x1) (binintersect x0 x2) leaving 2 subgoals.
The subproof is completed by applying H0.
Assume H1: x3 ∈ x0.
Assume H2: nIn x3 (setminus x1 x2).
Apply setminus_nIn_E with x1, x2, x3, x3 ∈ binunion (setminus x0 x1) (binintersect x0 x2) leaving 3 subgoals.
The subproof is completed by applying H2.
Assume H3: nIn x3 x1.
Apply binunionI1 with setminus x0 x1, binintersect x0 x2, x3.
Apply setminusI with x0, x1, x3 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
Assume H3: x3 ∈ x2.
Apply binunionI2 with setminus x0 x1, binintersect x0 x2, x3.
Apply binintersectI with x0, x2, x3 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
Apply binunion_Subq_min with setminus x0 x1, binintersect x0 x2, setminus x0 (setminus x1 x2) leaving 2 subgoals.
Let x3 of type ι be given.
Assume H0: x3 ∈ setminus x0 x1.
Apply setminusE with x0, x1, x3, x3 ∈ setminus x0 (setminus x1 x2) leaving 2 subgoals.
The subproof is completed by applying H0.
Assume H1: x3 ∈ x0.
Assume H2: nIn x3 x1.
Apply setminusI with x0, setminus x1 x2, x3 leaving 2 subgoals.
The subproof is completed by applying H1.
Apply setminus_nIn_I1 with x1, x2, x3.
The subproof is completed by applying H2.
Let x3 of type ι be given.
Assume H0: x3 ∈ binintersect x0 x2.
Apply binintersectE with x0, x2, x3, x3 ∈ setminus x0 (setminus x1 x2) leaving 2 subgoals.
The subproof is completed by applying H0.
Assume H1: x3 ∈ x0.
Assume H2: x3 ∈ x2.
Apply setminusI with x0, setminus x1 x2, x3 leaving 2 subgoals.
The subproof is completed by applying H1.
Apply setminus_nIn_I2 with x1, x2, x3.
The subproof is completed by applying H2.
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