Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax-rnegex Structured version   Visualization version   GIF version

Axiom ax-rnegex 10209
 Description: Existence of negative of real number. Axiom 15 of 22 for real and complex numbers, justified by theorem axrnegex 10185. (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 10137 . . 3 class
31, 2wcel 2145 . 2 wff 𝐴 ∈ ℝ
4 vx . . . . . 6 setvar 𝑥
54cv 1630 . . . . 5 class 𝑥
6 caddc 10141 . . . . 5 class +
71, 5, 6co 6793 . . . 4 class (𝐴 + 𝑥)
8 cc0 10138 . . . 4 class 0
97, 8wceq 1631 . . 3 wff (𝐴 + 𝑥) = 0
109, 4, 2wrex 3062 . 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  10242  00id  10413  addid1  10418  cnegex  10419  renegcli  10544
 Copyright terms: Public domain W3C validator