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Definition df-gbe 41961
Description: Define the set of (even) Goldbach numbers, which are positive even integers that can be expressed as the sum of two odd primes. By this definition, the binary Goldbach conjecture can be expressed as 𝑛 ∈ Even (4 < 𝑛𝑛 ∈ GoldbachEven ). (Contributed by AV, 14-Jun-2020.)
Assertion
Ref Expression
df-gbe GoldbachEven = {𝑧 ∈ Even ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))}
Distinct variable group:   𝑧,𝑝,𝑞

Detailed syntax breakdown of Definition df-gbe
StepHypRef Expression
1 cgbe 41958 . 2 class GoldbachEven
2 vp . . . . . . . 8 setvar 𝑝
32cv 1522 . . . . . . 7 class 𝑝
4 codd 41863 . . . . . . 7 class Odd
53, 4wcel 2030 . . . . . 6 wff 𝑝 ∈ Odd
6 vq . . . . . . . 8 setvar 𝑞
76cv 1522 . . . . . . 7 class 𝑞
87, 4wcel 2030 . . . . . 6 wff 𝑞 ∈ Odd
9 vz . . . . . . . 8 setvar 𝑧
109cv 1522 . . . . . . 7 class 𝑧
11 caddc 9977 . . . . . . . 8 class +
123, 7, 11co 6690 . . . . . . 7 class (𝑝 + 𝑞)
1310, 12wceq 1523 . . . . . 6 wff 𝑧 = (𝑝 + 𝑞)
145, 8, 13w3a 1054 . . . . 5 wff (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))
15 cprime 15432 . . . . 5 class
1614, 6, 15wrex 2942 . . . 4 wff 𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))
1716, 2, 15wrex 2942 . . 3 wff 𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))
18 ceven 41862 . . 3 class Even
1917, 9, 18crab 2945 . 2 class {𝑧 ∈ Even ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))}
201, 19wceq 1523 1 wff GoldbachEven = {𝑧 ∈ Even ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))}
Colors of variables: wff setvar class
This definition is referenced by:  isgbe  41964
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