Let x0 of type ι → (ι → ο) → ο be given.

Let x1 of type ι be given.

Assume H0: ordinal x1.

Let x2 of type ι → ο be given.

Let x3 of type ι be given.

Assume H1: prim1 x3 x1.

Apply H0 with dafc2.. x0 x1 x2 ⟶ dafc2.. x0 x3 x2.

Assume H2: TransSet x1.

Assume H3: ∀ x4 . prim1 x4 x1 ⟶ TransSet x4.

Claim L4: ordinal x3

Apply ordinal_Hered with x1, x3 leaving 2 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying H1.

Claim L5: TransSet x3

Apply L4 with TransSet x3.

Assume H5: TransSet x3.

Assume H6: ∀ x4 . prim1 x4 x3 ⟶ TransSet x4.

The subproof is completed by applying H5.

Assume H6: ∀ x4 . prim1 x4 x1 ⟶ ∀ x5 : ι → ο . 610d8.. x0 x4 x5 ⟶ 40dde.. x1 x2 x4 x5.

Let x4 of type ι be given.

Assume H7: prim1 x4 x3.

Let x5 of type ι → ο be given.

Assume H8: 610d8.. x0 x4 x5.

Claim L9: prim1 x4 x1

Apply H2 with x3, x4 leaving 2 subgoals.

The subproof is completed by applying H1.

The subproof is completed by applying H7.

Claim L10: 40dde.. x1 x2 x4 x5

Apply H6 with x4, x5 leaving 2 subgoals.

The subproof is completed by applying L9.

The subproof is completed by applying H8.

Apply unknownprop_1c12738cd89f8c615a541c15b6797bba2a5be97ab5e514c9fd76b3fef06e2aa9 with x1, x4, x2, x5, 40dde.. x3 x2 x4 x5 leaving 4 subgoals.

The subproof is completed by applying L10.

Assume H11: PNoLt_ (d3786.. x1 x4) x2 x5.

Claim L12: d3786.. x1 x4 = x4

Apply unknownprop_c3bea5de1408165b06631f86b9f132e6a4154b60456b567e9159f0db1e9656af with x1, x4, λ x6 x7 . x7 = x4.

Apply unknownprop_71c983f9883b4f28b7b9554926463bb4627a33e675a17eaba24400b257e59781 with x4, x1.

Apply H2 with x4.

The subproof is completed by applying L9.

Claim L13: d3786.. x3 x4 = x4

Apply unknownprop_c3bea5de1408165b06631f86b9f132e6a4154b60456b567e9159f0db1e9656af with x3, x4, λ x6 x7 . x7 = x4.

Apply unknownprop_71c983f9883b4f28b7b9554926463bb4627a33e675a17eaba24400b257e59781 with x4, x3.

Apply L5 with x4.

The subproof is completed by applying H7.

Apply unknownprop_ac970f51deca19d20e9a8350c3518ea802533ebd2768fe799c9b97ea3dd03596 with x3, x4, x2, x5.

Apply L13 with λ x6 x7 . PNoLt_ x7 x2 x5.

Apply L12 with λ x6 x7 . PNoLt_ x6 x2 x5.

The subproof is completed by applying H11.

Assume H11: prim1 x1 x4.

Apply FalseE with PNoEq_ x1 x2 x5 ⟶ x5 x1 ⟶ 40dde.. x3 x2 x4 x5.

Apply unknownprop_f1a526a64fd91875cd825eea7f2e7776b7f0e7be5dcee74dc03af1d7886d1eb6 with x1, x4 leaving 2 subgoals.

The subproof is completed by applying H11.

The subproof is completed by applying L9.

Assume H11: prim1 x4 x1.

Assume H12: PNoEq_ x4 x2 x5.

Assume H13: not (x2 x4).

Apply unknownprop_b51c738b3a14385af55eef02a445728dc056a37996fdc42e5ede8e064af23c97 with x3, x4, x2, x5 leaving 3 subgoals.

The subproof is completed by applying H7.

The subproof is completed by applying H12.

The subproof is completed by applying H13.

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Proofgold Proof