Definition df-gbe 46416

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 46413	. 2 class GoldbachEven
2		vp	. . . . . . . 8 setvar 𝑝
3	2	cv 1541	. . . . . . 7 class 𝑝
4		codd 46293	. . . . . . 7 class Odd
5	3, 4	wcel 2107	. . . . . 6 wff 𝑝 ∈ Odd
6		vq	. . . . . . . 8 setvar 𝑞
7	6	cv 1541	. . . . . . 7 class 𝑞
8	7, 4	wcel 2107	. . . . . 6 wff 𝑞 ∈ Odd
9		vz	. . . . . . . 8 setvar 𝑧
10	9	cv 1541	. . . . . . 7 class 𝑧
11		caddc 11113	. . . . . . . 8 class +
12	3, 7, 11	co 7409	. . . . . . 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 16608	. . . . 5 class ℙ
16	14, 6, 15	wrex 3071	. . . 4 wff ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))
17	16, 2, 15	wrex 3071	. . 3 wff ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))
18		ceven 46292	. . 3 class Even
19	17, 9, 18	crab 3433	. 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  46419