Definition df-gbe 47936

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 47933	. 2 class GoldbachEven
2		vp	. . . . . . . 8 setvar 𝑝
3	2	cv 1540	. . . . . . 7 class 𝑝
4		codd 47813	. . . . . . 7 class Odd
5	3, 4	wcel 2113	. . . . . 6 wff 𝑝 ∈ Odd
6		vq	. . . . . . . 8 setvar 𝑞
7	6	cv 1540	. . . . . . 7 class 𝑞
8	7, 4	wcel 2113	. . . . . 6 wff 𝑞 ∈ Odd
9		vz	. . . . . . . 8 setvar 𝑧
10	9	cv 1540	. . . . . . 7 class 𝑧
11		caddc 11027	. . . . . . . 8 class +
12	3, 7, 11	co 7356	. . . . . . 7 class (𝑝 + 𝑞)
13	10, 12	wceq 1541	. . . . . 6 wff 𝑧 = (𝑝 + 𝑞)
14	5, 8, 13	w3a 1086	. . . . 5 wff (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))
15		cprime 16596	. . . . 5 class ℙ
16	14, 6, 15	wrex 3058	. . . 4 wff ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))
17	16, 2, 15	wrex 3058	. . 3 wff ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))
18		ceven 47812	. . 3 class Even
19	17, 9, 18	crab 3397	. 2 class {𝑧 ∈ Even ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))}
20	1, 19	wceq 1541	1 wff GoldbachEven = {𝑧 ∈ Even ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))}

This definition is referenced by:  isgbe  47939