Definition df-gbe 46714

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 46711	. 2 class GoldbachEven
2		vp	. . . . . . . 8 setvar 𝑝
3	2	cv 1538	. . . . . . 7 class 𝑝
4		codd 46591	. . . . . . 7 class Odd
5	3, 4	wcel 2104	. . . . . 6 wff 𝑝 ∈ Odd
6		vq	. . . . . . . 8 setvar 𝑞
7	6	cv 1538	. . . . . . 7 class 𝑞
8	7, 4	wcel 2104	. . . . . 6 wff 𝑞 ∈ Odd
9		vz	. . . . . . . 8 setvar 𝑧
10	9	cv 1538	. . . . . . 7 class 𝑧
11		caddc 11115	. . . . . . . 8 class +
12	3, 7, 11	co 7411	. . . . . . 7 class (𝑝 + 𝑞)
13	10, 12	wceq 1539	. . . . . 6 wff 𝑧 = (𝑝 + 𝑞)
14	5, 8, 13	w3a 1085	. . . . 5 wff (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))
15		cprime 16612	. . . . 5 class ℙ
16	14, 6, 15	wrex 3068	. . . 4 wff ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))
17	16, 2, 15	wrex 3068	. . 3 wff ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))
18		ceven 46590	. . 3 class Even
19	17, 9, 18	crab 3430	. 2 class {𝑧 ∈ Even ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))}
20	1, 19	wceq 1539	1 wff GoldbachEven = {𝑧 ∈ Even ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))}

This definition is referenced by:  isgbe  46717