Definition df-gbe 43912

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 < 𝑛 → 𝑛 ∈ 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

Step	Hyp	Ref	Expression
1		cgbe 43909	. 2 class GoldbachEven
2		vp	. . . . . . . 8 setvar 𝑝
3	2	cv 1532	. . . . . . 7 class 𝑝
4		codd 43789	. . . . . . 7 class Odd
5	3, 4	wcel 2110	. . . . . 6 wff 𝑝 ∈ Odd
6		vq	. . . . . . . 8 setvar 𝑞
7	6	cv 1532	. . . . . . 7 class 𝑞
8	7, 4	wcel 2110	. . . . . 6 wff 𝑞 ∈ Odd
9		vz	. . . . . . . 8 setvar 𝑧
10	9	cv 1532	. . . . . . 7 class 𝑧
11		caddc 10539	. . . . . . . 8 class +
12	3, 7, 11	co 7155	. . . . . . 7 class (𝑝 + 𝑞)
13	10, 12	wceq 1533	. . . . . 6 wff 𝑧 = (𝑝 + 𝑞)
14	5, 8, 13	w3a 1083	. . . . 5 wff (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))
15		cprime 16014	. . . . 5 class ℙ
16	14, 6, 15	wrex 3139	. . . 4 wff ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))
17	16, 2, 15	wrex 3139	. . 3 wff ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))
18		ceven 43788	. . 3 class Even
19	17, 9, 18	crab 3142	. 2 class {𝑧 ∈ Even ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))}
20	1, 19	wceq 1533	1 wff GoldbachEven = {𝑧 ∈ Even ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))}

This definition is referenced by:  isgbe  43915