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Definition df-gbe 45200
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 45197 . 2 class GoldbachEven
2 vp . . . . . . . 8 setvar 𝑝
32cv 1538 . . . . . . 7 class 𝑝
4 codd 45077 . . . . . . 7 class Odd
53, 4wcel 2106 . . . . . 6 wff 𝑝 ∈ Odd
6 vq . . . . . . . 8 setvar 𝑞
76cv 1538 . . . . . . 7 class 𝑞
87, 4wcel 2106 . . . . . 6 wff 𝑞 ∈ Odd
9 vz . . . . . . . 8 setvar 𝑧
109cv 1538 . . . . . . 7 class 𝑧
11 caddc 10874 . . . . . . . 8 class +
123, 7, 11co 7275 . . . . . . 7 class (𝑝 + 𝑞)
1310, 12wceq 1539 . . . . . 6 wff 𝑧 = (𝑝 + 𝑞)
145, 8, 13w3a 1086 . . . . 5 wff (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))
15 cprime 16376 . . . . 5 class
1614, 6, 15wrex 3065 . . . 4 wff 𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))
1716, 2, 15wrex 3065 . . 3 wff 𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))
18 ceven 45076 . . 3 class Even
1917, 9, 18crab 3068 . 2 class {𝑧 ∈ Even ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))}
201, 19wceq 1539 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  45203
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