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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: ordinal x0.
Let x1 of type ιο be given.
Assume H1: ∀ x2 . x2x0∀ x3 . x3x2x1 x2x1 x3.
Apply andI with TransSet {x2 ∈ x0|x1 x2}, ∀ x2 . x2{x3 ∈ x0|x1 x3}TransSet x2 leaving 2 subgoals.
Let x2 of type ι be given.
Assume H2: x2{x3 ∈ x0|x1 x3}.
Let x3 of type ι be given.
Assume H3: x3x2.
Apply SepE with x0, x1, x2, x3{x4 ∈ x0|x1 x4} leaving 2 subgoals.
The subproof is completed by applying H2.
Assume H4: x2x0.
Assume H5: x1 x2.
Apply SepI with x0, x1, x3 leaving 2 subgoals.
Apply H0 with x3x0.
Assume H6: TransSet x0.
Assume H7: ∀ x4 . x4x0TransSet x4.
Apply H6 with x2, x3 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H3.
Apply H1 with x2, x3 leaving 3 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H3.
The subproof is completed by applying H5.
Let x2 of type ι be given.
Assume H2: x2{x3 ∈ x0|x1 x3}.
Claim L3: x2x0
Apply SepE1 with x0, x1, x2.
The subproof is completed by applying H2.
Apply ordinal_Hered with x0, x2, TransSet x2 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L3.
Assume H4: TransSet x2.
Assume H5: ∀ x3 . x3x2TransSet x3.
The subproof is completed by applying H4.