![]() |
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 11221 without using ax-1cn 11170 and ax-cnre 11185. (Contributed by Steven Nguyen, 28-Jan-2023.) |
Ref | Expression |
---|---|
elre0re | ⊢ (𝐴 ∈ ℝ → 0 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-rnegex 11183 | . 2 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) | |
2 | readdcl 11195 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝐴 + 𝑥) ∈ ℝ) | |
3 | eleq1 2821 | . . . 4 ⊢ ((𝐴 + 𝑥) = 0 → ((𝐴 + 𝑥) ∈ ℝ ↔ 0 ∈ ℝ)) | |
4 | 2, 3 | syl5ibcom 244 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝐴 + 𝑥) = 0 → 0 ∈ ℝ)) |
5 | 4 | rexlimdva 3155 | . 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 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3070 (class class class)co 7411 ℝcr 11111 0cc0 11112 + caddc 11115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-addrcl 11173 ax-rnegex 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 df-cleq 2724 df-clel 2810 df-rex 3071 |
This theorem is referenced by: rernegcl 41546 renegadd 41547 reneg0addlid 41549 resubeulem1 41550 resubeulem2 41551 resubeu 41552 remul02 41580 remul01 41582 readdrid 41584 resubid1 41585 renegneg 41586 renegid2 41588 sn-it0e0 41590 relt0neg2 41620 |
Copyright terms: Public domain | W3C validator |