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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: ordinal x0.
Let x1 of type ι be given.
Assume H1: ordinal x1.
Claim L2: ...
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Claim L3: ...
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Claim L4: ...
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Claim L5: ...
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Apply SNo_max_ordinal with add_SNo x0 x1 leaving 2 subgoals.
The subproof is completed by applying L4.
Let x2 of type ι be given.
Assume H6: x2 ∈ SNoS_ (SNoLev (add_SNo x0 x1)).
Apply SNoS_E2 with SNoLev (add_SNo x0 x1), x2, SNoLt x2 (add_SNo x0 x1) leaving 3 subgoals.
The subproof is completed by applying L5.
The subproof is completed by applying H6.
Assume H7: SNoLev x2 ∈ SNoLev (add_SNo x0 x1).
Assume H8: ordinal (SNoLev x2).
Assume H9: SNo x2.
Assume H10: SNo_ (SNoLev x2) x2.
Apply SNoLt_trichotomy_or with x2, add_SNo x0 x1, SNoLt x2 (add_SNo x0 x1) leaving 4 subgoals.
The subproof is completed by applying H9.
The subproof is completed by applying L4.
Assume H11: or (SNoLt x2 (add_SNo x0 x1)) (x2 = add_SNo x0 x1).
Apply H11 with SNoLt x2 (add_SNo x0 x1) leaving 2 subgoals.
Assume H12: SNoLt x2 (add_SNo x0 x1).
The subproof is completed by applying H12.
Assume H12: x2 = add_SNo x0 x1.
Apply FalseE with SNoLt x2 (add_SNo x0 x1).
Apply In_irref with SNoLev x2.
Apply H12 with λ x3 x4 . SNoLev x2 ∈ SNoLev x4.
The subproof is completed by applying H7.
Assume H11: SNoLt (add_SNo x0 x1) x2.
Apply FalseE with SNoLt x2 (add_SNo x0 x1).
Apply add_SNo_ordinal_SNoCutP with x0, x1, False leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Assume H12: and (∀ x3 . x3 ∈ binunion {add_SNo x4 x1|x4 ∈ SNoS_ x0} (prim5 (SNoS_ x1) (add_SNo x0)) ⟶ SNo x3) (∀ x3 . x3 ∈ 0 ⟶ SNo x3).
Apply H12 with (∀ x3 . x3 ∈ binunion {add_SNo x4 x1|x4 ∈ SNoS_ x0} (prim5 (SNoS_ x1) (add_SNo x0)) ⟶ ∀ x4 . x4 ∈ 0 ⟶ SNoLt x3 x4) ⟶ False.
Assume H13: ∀ x3 . x3 ∈ binunion {add_SNo x4 x1|x4 ∈ SNoS_ x0} {add_SNo x0 x4|x4 ∈ SNoS_ x1} ⟶ SNo x3.
Assume H14: ∀ x3 . x3 ∈ 0 ⟶ SNo x3.
Assume H15: ∀ x3 . x3 ∈ binunion {add_SNo x4 x1|x4 ∈ SNoS_ x0} (prim5 (SNoS_ x1) (add_SNo x0)) ⟶ ∀ x4 . x4 ∈ 0 ⟶ SNoLt x3 x4.
Apply SNoCutP_SNoCut with binunion {add_SNo x3 x1|x3 ∈ SNoS_ x0} {add_SNo x0 x3|x3 ∈ SNoS_ x1}, 0, False leaving 2 subgoals.
Apply add_SNo_ordinal_SNoCutP with x0, x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Assume H16: and (and (and (SNo (SNoCut (binunion {add_SNo x3 x1|x3 ∈ SNoS_ x0} (prim5 (SNoS_ x1) (add_SNo x0))) 0)) (SNoLev (SNoCut (binunion {add_SNo x3 x1|x3 ∈ SNoS_ x0} (prim5 (SNoS_ x1) (add_SNo x0))) 0) ∈ ordsucc (binunion (famunion (binunion {...|x3 ∈ ...} ...) ...) ...))) ...) ....
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