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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H0: ordinal x0.
Assume H1: ordinal x1.
Apply H0 with ordinal (binunion x0 x1).
Assume H2: TransSet x0.
Assume H3: ∀ x2 . x2x0TransSet x2.
Apply H1 with ordinal (binunion x0 x1).
Assume H4: TransSet x1.
Assume H5: ∀ x2 . x2x1TransSet x2.
Apply ordinal_linear with x0, x1, ordinal (binunion x0 x1) leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply Subq_binunion_eq with x0, x1, λ x2 x3 : ο . x3ordinal (binunion x0 x1).
Assume H6: binunion x0 x1 = x1.
Apply H6 with λ x2 x3 . ordinal x3.
The subproof is completed by applying H1.
Apply Subq_binunion_eq with x1, x0, λ x2 x3 : ο . x3ordinal (binunion x0 x1).
Apply binunion_com with x1, x0, λ x2 x3 . x3 = x0ordinal (binunion x0 x1).
Assume H6: binunion x0 x1 = x0.
Apply H6 with λ x2 x3 . ordinal x3.
The subproof is completed by applying H0.