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Axiom ax-rnegex 11226
Description: Existence of negative of real number. Axiom 15 of 22 for real and complex numbers, justified by Theorem axrnegex 11202. (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
StepHypRef Expression
1 cA . . 3 class 𝐴
2 cr 11154 . . 3 class
31, 2wcel 2108 . 2 wff 𝐴 ∈ ℝ
4 vx . . . . . 6 setvar 𝑥
54cv 1539 . . . . 5 class 𝑥
6 caddc 11158 . . . . 5 class +
71, 5, 6co 7431 . . . 4 class (𝐴 + 𝑥)
8 cc0 11155 . . . 4 class 0
97, 8wceq 1540 . . 3 wff (𝐴 + 𝑥) = 0
109, 4, 2wrex 3070 . 2 wff 𝑥 ∈ ℝ (𝐴 + 𝑥) = 0
113, 10wi 4 1 wff (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
Colors of variables: wff setvar class
This axiom is referenced by:  0re  11263  00id  11436  addrid  11441  cnegex  11442  0cnALT  11496  renegcli  11570  elre0re  42295  renegeu  42400
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