Axiom ax-rnegex 11177

Description: Existence of negative of real number. Axiom 15 of 22 for real and complex numbers, justified by Theorem axrnegex 11153. (Contributed by Eric Schmidt, 21-May-2007.)

Assertion

Ref	Expression
ax-rnegex	⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)

Distinct variable group:   𝑥,𝐴

Detailed syntax breakdown of Axiom ax-rnegex

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		cr 11105	. . 3 class ℝ
3	1, 2	wcel 2106	. 2 wff 𝐴 ∈ ℝ
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1540	. . . . 5 class 𝑥
6		caddc 11109	. . . . 5 class +
7	1, 5, 6	co 7405	. . . 4 class (𝐴 + 𝑥)
8		cc0 11106	. . . 4 class 0
9	7, 8	wceq 1541	. . 3 wff (𝐴 + 𝑥) = 0
10	9, 4, 2	wrex 3070	. 2 wff ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0
11	3, 10	wi 4	1 wff (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)

This axiom is referenced by:  0re  11212  00id  11385  addrid  11390  cnegex  11391  0cnALT  11444  renegcli  11517  elre0re  41172  renegeu  41239