Mathbox for Steven Nguyen |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > elre0re | Structured version Visualization version GIF version |
Description: Specialized version of 0red 10646 without using ax-1cn 10597 and ax-cnre 10612. (Contributed by Steven Nguyen, 28-Jan-2023.) |
Ref | Expression |
---|---|
elre0re | ⊢ (𝐴 ∈ ℝ → 0 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-rnegex 10610 | . 2 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) | |
2 | readdcl 10622 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝐴 + 𝑥) ∈ ℝ) | |
3 | eleq1 2902 | . . . 4 ⊢ ((𝐴 + 𝑥) = 0 → ((𝐴 + 𝑥) ∈ ℝ ↔ 0 ∈ ℝ)) | |
4 | 2, 3 | syl5ibcom 247 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝐴 + 𝑥) = 0 → 0 ∈ ℝ)) |
5 | 4 | rexlimdva 3286 | . 2 ⊢ (𝐴 ∈ ℝ → (∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0 → 0 ∈ ℝ)) |
6 | 1, 5 | mpd 15 | 1 ⊢ (𝐴 ∈ ℝ → 0 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 (class class class)co 7158 ℝcr 10538 0cc0 10539 + caddc 10542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-ext 2795 ax-addrcl 10600 ax-rnegex 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-cleq 2816 df-clel 2895 df-ral 3145 df-rex 3146 |
This theorem is referenced by: rernegcl 39208 renegadd 39209 reneg0addid2 39211 resubeulem1 39212 resubeulem2 39213 resubeu 39214 remul02 39242 remul01 39244 readdid1 39246 resubid1 39247 renegneg 39248 relt0neg1 39251 relt0neg2 39252 |
Copyright terms: Public domain | W3C validator |