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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.
Assume H0: SNo_min_of x0 x1.
Assume H1: SNo_min_of x0 x2.
Apply H0 with x1 = x2.
Assume H2: and (x1x0) (SNo x1).
Apply H2 with (∀ x3 . x3x0SNo x3SNoLe x1 x3)x1 = x2.
Assume H3: x1x0.
Assume H4: SNo x1.
Assume H5: ∀ x3 . x3x0SNo x3SNoLe x1 x3.
Apply H1 with x1 = x2.
Assume H6: and (x2x0) (SNo x2).
Apply H6 with (∀ x3 . x3x0SNo x3SNoLe x2 x3)x1 = x2.
Assume H7: x2x0.
Assume H8: SNo x2.
Assume H9: ∀ x3 . x3x0SNo x3SNoLe x2 x3.
Apply SNoLe_antisym with x1, x2 leaving 4 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H8.
Apply H5 with x2 leaving 2 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
Apply H9 with x1 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.