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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 ιιο be given.
Assume H0: ∀ x3 . x3x1∀ x4 . x4x1x2 x3 x4x2 x4 x3.
Assume H1: 4402e.. x1 x2.
Let x3 of type ι be given.
Assume H2: x3x1.
Assume H3: x0setminus x1 (Sing x3).
Let x4 of type ι be given.
Assume H4: x4x0.
Let x5 of type ι be given.
Assume H5: x5x0.
Let x6 of type ι be given.
Assume H6: x6x0.
Let x7 of type ι be given.
Assume H7: x7x0.
Apply setminusE with x1, Sing x3, x4, 2f869.. x2 x4 x5 x6 x7∀ x8 : ο . (∀ x9 . x9x0∀ x10 . x10x0∀ x11 . x11x0∀ x12 . x12x05a3b5.. x2 x9 x10 x11 x12 x3x8)(∀ x9 . x9x0∀ x10 . x10x0∀ x11 . x11x0∀ x12 . x12x06ff44.. x2 x9 x10 x11 x12 x3x8)(∀ x9 . x9x0∀ x10 . x10x0∀ x11 . x11x0∀ x12 . x12x062523.. x2 x9 x10 x3 x11 x12x8)(∀ x9 . x9x0∀ x10 . x10x0∀ x11 . x11x0∀ x12 . x12x087c36.. x2 x9 x10 x11 x12 x3x8)(∀ x9 . x9x0∀ x10 . x10x0∀ x11 . x11x0∀ x12 . x12x042d13.. x2 x9 x10 x11 x12 x3x8)(∀ x9 . x9x0∀ x10 . x10x0∀ x11 . x11x0∀ x12 . x12x0c5756.. x2 x9 x10 x11 x3 x12x8)x8 leaving 2 subgoals.
Apply H3 with x4.
The subproof is completed by applying H4.
Assume H8: x4x1.
Assume H9: nIn x4 (Sing x3).
Apply setminusE with x1, Sing x3, x5, ...∀ x8 : ο . .........(∀ x9 . ...∀ x10 . ...∀ x11 . ...∀ x12 . ...87c36.. x2 x9 x10 x11 ... ...x8)(∀ x9 . x9x0∀ x10 . x10x0∀ x11 . x11x0∀ x12 . x12x042d13.. x2 x9 x10 x11 x12 x3x8)(∀ x9 . x9x0∀ x10 . x10x0∀ x11 . x11x0∀ x12 . x12x0c5756.. x2 x9 x10 x11 x3 x12x8)x8 leaving 2 subgoals.
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