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