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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: nat_p x0.
Apply nat_inv with x0, mul_nat 2 x0 ⊆ x0 ⟶ x0 = 0 leaving 3 subgoals.
The subproof is completed by applying H0.
Assume H1: x0 = 0.
Assume H2: mul_nat 2 x0 ⊆ x0.
The subproof is completed by applying H1.
Assume H1: ∃ x1 . and (nat_p x1) (x0 = ordsucc x1).
Apply H1 with mul_nat 2 x0 ⊆ x0 ⟶ x0 = 0.
Let x1 of type ι be given.
Assume H2: (λ x2 . and (nat_p x2) (x0 = ordsucc x2)) x1.
Apply H2 with mul_nat 2 x0 ⊆ x0 ⟶ x0 = 0.
Assume H3: nat_p x1.
Assume H4: x0 = ordsucc x1.
Apply H4 with λ x2 x3 . mul_nat 2 x3 ⊆ x3 ⟶ x3 = 0.
Assume H5: mul_nat 2 (ordsucc x1) ⊆ ordsucc x1.
Apply FalseE with ordsucc x1 = 0.
Apply In_irref with ordsucc x1.
Apply H5 with ordsucc x1.
Apply ordsucc_in_double_nat_ordsucc with x1.
The subproof is completed by applying H3.
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