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 < 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 44761	. 2 class GoldbachEven
2		vp	. . . . . . . 8 setvar 𝑝
3	2	cv 1541	. . . . . . 7 class 𝑝
4		codd 44641	. . . . . . 7 class Odd
5	3, 4	wcel 2114	. . . . . 6 wff 𝑝 ∈ Odd
6		vq	. . . . . . . 8 setvar 𝑞
7	6	cv 1541	. . . . . . 7 class 𝑞
8	7, 4	wcel 2114	. . . . . 6 wff 𝑞 ∈ Odd
9		vz	. . . . . . . 8 setvar 𝑧
10	9	cv 1541	. . . . . . 7 class 𝑧
11		caddc 10621	. . . . . . . 8 class +
12	3, 7, 11	co 7173	. . . . . . 7 class (𝑝 + 𝑞)
13	10, 12	wceq 1542	. . . . . 6 wff 𝑧 = (𝑝 + 𝑞)
14	5, 8, 13	w3a 1088	. . . . 5 wff (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))
15		cprime 16115	. . . . 5 class ℙ
16	14, 6, 15	wrex 3055	. . . 4 wff ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))
17	16, 2, 15	wrex 3055	. . 3 wff ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))
18		ceven 44640	. . 3 class Even
19	17, 9, 18	crab 3058	. 2 class {𝑧 ∈ Even ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))}
20	1, 19	wceq 1542	1 wff GoldbachEven = {𝑧 ∈ Even ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))}

This definition is referenced by:  isgbe  44767