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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H0: SNo x0.
Assume H1: SNo x1.
Apply PNoLt_trichotomy_or with SNoLev x0, SNoLev x1, λ x2 . x2 ∈ x0, λ x2 . x2 ∈ x1, or (or (SNoLt x0 x1) (x0 = x1)) (SNoLt x1 x0) leaving 4 subgoals.
Apply SNoLev_ordinal with x0.
The subproof is completed by applying H0.
Apply SNoLev_ordinal with x1.
The subproof is completed by applying H1.
Assume H2: or (PNoLt (SNoLev x0) (λ x2 . x2 ∈ x0) (SNoLev x1) (λ x2 . x2 ∈ x1)) (and (SNoLev x0 = SNoLev x1) (PNoEq_ (SNoLev x0) (λ x2 . x2 ∈ x0) (λ x2 . x2 ∈ x1))).
Apply H2 with or (or (SNoLt x0 x1) (x0 = x1)) (SNoLt x1 x0) leaving 2 subgoals.
Assume H3: PNoLt (SNoLev x0) (λ x2 . x2 ∈ x0) (SNoLev x1) (λ x2 . x2 ∈ x1).
Apply or3I1 with SNoLt x0 x1, x0 = x1, SNoLt x1 x0.
The subproof is completed by applying H3.
Assume H3: and (SNoLev x0 = SNoLev x1) (PNoEq_ (SNoLev x0) (λ x2 . x2 ∈ x0) (λ x2 . x2 ∈ x1)).
Apply H3 with or (or (SNoLt x0 x1) (x0 = x1)) (SNoLt x1 x0).
Assume H4: SNoLev x0 = SNoLev x1.
Assume H5: PNoEq_ (SNoLev x0) (λ x2 . x2 ∈ x0) (λ x2 . x2 ∈ x1).
Apply or3I2 with SNoLt x0 x1, x0 = x1, SNoLt x1 x0.
Apply SNo_eq with x0, x1 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
Assume H2: PNoLt (SNoLev x1) (λ x2 . x2 ∈ x1) (SNoLev x0) (λ x2 . x2 ∈ x0).
Apply or3I3 with SNoLt x0 x1, x0 = x1, SNoLt x1 x0.
The subproof is completed by applying H2.
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