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Definition df-gbe 44764
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 44761 . 2 class GoldbachEven
2 vp . . . . . . . 8 setvar 𝑝
32cv 1541 . . . . . . 7 class 𝑝
4 codd 44641 . . . . . . 7 class Odd
53, 4wcel 2114 . . . . . 6 wff 𝑝 ∈ Odd
6 vq . . . . . . . 8 setvar 𝑞
76cv 1541 . . . . . . 7 class 𝑞
87, 4wcel 2114 . . . . . 6 wff 𝑞 ∈ Odd
9 vz . . . . . . . 8 setvar 𝑧
109cv 1541 . . . . . . 7 class 𝑧
11 caddc 10621 . . . . . . . 8 class +
123, 7, 11co 7173 . . . . . . 7 class (𝑝 + 𝑞)
1310, 12wceq 1542 . . . . . 6 wff 𝑧 = (𝑝 + 𝑞)
145, 8, 13w3a 1088 . . . . 5 wff (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))
15 cprime 16115 . . . . 5 class
1614, 6, 15wrex 3055 . . . 4 wff 𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))
1716, 2, 15wrex 3055 . . 3 wff 𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))
18 ceven 44640 . . 3 class Even
1917, 9, 18crab 3058 . 2 class {𝑧 ∈ Even ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))}
201, 19wceq 1542 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  44767
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