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| Mirrors > Home > MPE Home > Th. List > ax-rnegex | Structured version Visualization version GIF version | ||
| 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.) |
| Ref | Expression |
|---|---|
| ax-rnegex | ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cA | . . 3 class 𝐴 | |
| 2 | cr 11154 | . . 3 class ℝ | |
| 3 | 1, 2 | wcel 2108 | . 2 wff 𝐴 ∈ ℝ |
| 4 | vx | . . . . . 6 setvar 𝑥 | |
| 5 | 4 | cv 1539 | . . . . 5 class 𝑥 |
| 6 | caddc 11158 | . . . . 5 class + | |
| 7 | 1, 5, 6 | co 7431 | . . . 4 class (𝐴 + 𝑥) |
| 8 | cc0 11155 | . . . 4 class 0 | |
| 9 | 7, 8 | wceq 1540 | . . 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) |
| 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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