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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 (binunion x1 x2), setminus (setminus x0 x1) x2 leaving 2 subgoals.
Let x3 of type ι be given.
Assume H0: x3 ∈ setminus x0 (binunion x1 x2).
Apply setminusE with x0, binunion x1 x2, x3, x3 ∈ setminus (setminus x0 x1) x2 leaving 2 subgoals.
The subproof is completed by applying H0.
Assume H1: x3 ∈ x0.
Assume H2: nIn x3 (binunion x1 x2).
Apply binunion_nIn_E with x1, x2, x3, x3 ∈ setminus (setminus x0 x1) x2 leaving 2 subgoals.
The subproof is completed by applying H2.
Assume H3: nIn x3 x1.
Assume H4: nIn x3 x2.
Apply setminusI with setminus x0 x1, x2, x3 leaving 2 subgoals.
Apply setminusI with x0, x1, x3 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
Let x3 of type ι be given.
Assume H0: x3 ∈ setminus (setminus x0 x1) x2.
Apply setminusE with setminus x0 x1, x2, x3, x3 ∈ setminus x0 (binunion x1 x2) leaving 2 subgoals.
The subproof is completed by applying H0.
Assume H1: x3 ∈ setminus x0 x1.
Assume H2: nIn x3 x2.
Apply setminusE with x0, x1, x3, x3 ∈ setminus x0 (binunion x1 x2) leaving 2 subgoals.
The subproof is completed by applying H1.
Assume H3: x3 ∈ x0.
Assume H4: nIn x3 x1.
Apply setminusI with x0, binunion x1 x2, x3 leaving 2 subgoals.
The subproof is completed by applying H3.
Apply binunion_nIn_I with x1, x2, x3 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H2.
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