Assume Heq: setprod 2 ω = (ω ∖ {0}).
We will prove False.
We prove the intermediate claim H1omega: 1 ∈ ω.
An exact proof term for the current goal is (nat_p_omega 1 nat_1).
We prove the intermediate claim Hp: (0,1) ∈ setprod 2 ω.
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma 2 ω 0 1 In_0_2 H1omega).
We prove the intermediate claim HpNZ: (0,1) ∈ (ω ∖ {0}).
rewrite the current goal using Heq (from right to left).
An exact proof term for the current goal is Hp.
We prove the intermediate claim Hcore: (0,1) ∈ ω ∧ (0,1) ∉ {0}.
An exact proof term for the current goal is (setminusE ω {0} (0,1) HpNZ).
We prove the intermediate claim HpOmega: (0,1) ∈ ω.
An exact proof term for the current goal is (andEL ((0,1) ∈ ω) ((0,1) ∉ {0}) Hcore).
We prove the intermediate claim HSingOmega: {1} ∈ ω.
rewrite the current goal using tuple_0_1_eq_Sing1 (from right to left).
An exact proof term for the current goal is HpOmega.
An exact proof term for the current goal is (Sing1_not_in_omega HSingOmega).
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