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Mirrors > Home > ILE Home > Th. List > elrealeu | GIF version |
Description: The real number mapping in elreal 7557 is unique. (Contributed by Jim Kingdon, 11-Jul-2021.) |
Ref | Expression |
---|---|
elrealeu | ⊢ (𝐴 ∈ ℝ ↔ ∃!𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elreal 7557 | . . . 4 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | |
2 | 1 | biimpi 119 | . . 3 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) |
3 | eqtr3 2132 | . . . . . . . 8 ⊢ ((〈𝑥, 0R〉 = 𝐴 ∧ 〈𝑦, 0R〉 = 𝐴) → 〈𝑥, 0R〉 = 〈𝑦, 0R〉) | |
4 | 0r 7487 | . . . . . . . . . 10 ⊢ 0R ∈ R | |
5 | opthg 4118 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ R ∧ 0R ∈ R) → (〈𝑥, 0R〉 = 〈𝑦, 0R〉 ↔ (𝑥 = 𝑦 ∧ 0R = 0R))) | |
6 | 4, 5 | mpan2 419 | . . . . . . . . 9 ⊢ (𝑥 ∈ R → (〈𝑥, 0R〉 = 〈𝑦, 0R〉 ↔ (𝑥 = 𝑦 ∧ 0R = 0R))) |
7 | 6 | ad2antlr 478 | . . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ R) ∧ 𝑦 ∈ R) → (〈𝑥, 0R〉 = 〈𝑦, 0R〉 ↔ (𝑥 = 𝑦 ∧ 0R = 0R))) |
8 | 3, 7 | syl5ib 153 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ R) ∧ 𝑦 ∈ R) → ((〈𝑥, 0R〉 = 𝐴 ∧ 〈𝑦, 0R〉 = 𝐴) → (𝑥 = 𝑦 ∧ 0R = 0R))) |
9 | simpl 108 | . . . . . . 7 ⊢ ((𝑥 = 𝑦 ∧ 0R = 0R) → 𝑥 = 𝑦) | |
10 | 8, 9 | syl6 33 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ R) ∧ 𝑦 ∈ R) → ((〈𝑥, 0R〉 = 𝐴 ∧ 〈𝑦, 0R〉 = 𝐴) → 𝑥 = 𝑦)) |
11 | 10 | ralrimiva 2477 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ R) → ∀𝑦 ∈ R ((〈𝑥, 0R〉 = 𝐴 ∧ 〈𝑦, 0R〉 = 𝐴) → 𝑥 = 𝑦)) |
12 | 11 | ralrimiva 2477 | . . . 4 ⊢ (𝐴 ∈ ℝ → ∀𝑥 ∈ R ∀𝑦 ∈ R ((〈𝑥, 0R〉 = 𝐴 ∧ 〈𝑦, 0R〉 = 𝐴) → 𝑥 = 𝑦)) |
13 | opeq1 3669 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 〈𝑥, 0R〉 = 〈𝑦, 0R〉) | |
14 | 13 | eqeq1d 2121 | . . . . 5 ⊢ (𝑥 = 𝑦 → (〈𝑥, 0R〉 = 𝐴 ↔ 〈𝑦, 0R〉 = 𝐴)) |
15 | 14 | rmo4 2844 | . . . 4 ⊢ (∃*𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴 ↔ ∀𝑥 ∈ R ∀𝑦 ∈ R ((〈𝑥, 0R〉 = 𝐴 ∧ 〈𝑦, 0R〉 = 𝐴) → 𝑥 = 𝑦)) |
16 | 12, 15 | sylibr 133 | . . 3 ⊢ (𝐴 ∈ ℝ → ∃*𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) |
17 | reu5 2615 | . . 3 ⊢ (∃!𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴 ↔ (∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴 ∧ ∃*𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴)) | |
18 | 2, 16, 17 | sylanbrc 411 | . 2 ⊢ (𝐴 ∈ ℝ → ∃!𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) |
19 | reurex 2616 | . . 3 ⊢ (∃!𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴 → ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | |
20 | 19, 1 | sylibr 133 | . 2 ⊢ (∃!𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴 → 𝐴 ∈ ℝ) |
21 | 18, 20 | impbii 125 | 1 ⊢ (𝐴 ∈ ℝ ↔ ∃!𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1312 ∈ wcel 1461 ∀wral 2388 ∃wrex 2389 ∃!wreu 2390 ∃*wrmo 2391 〈cop 3494 Rcnr 7047 0Rc0r 7048 ℝcr 7540 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-coll 4001 ax-sep 4004 ax-nul 4012 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-iinf 4460 |
This theorem depends on definitions: df-bi 116 df-dc 803 df-3or 944 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-ral 2393 df-rex 2394 df-reu 2395 df-rmo 2396 df-rab 2397 df-v 2657 df-sbc 2877 df-csb 2970 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-nul 3328 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-int 3736 df-iun 3779 df-br 3894 df-opab 3948 df-mpt 3949 df-tr 3985 df-eprel 4169 df-id 4173 df-po 4176 df-iso 4177 df-iord 4246 df-on 4248 df-suc 4251 df-iom 4463 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-f1 5084 df-fo 5085 df-f1o 5086 df-fv 5087 df-ov 5729 df-oprab 5730 df-mpo 5731 df-1st 5990 df-2nd 5991 df-recs 6154 df-irdg 6219 df-1o 6265 df-oadd 6269 df-omul 6270 df-er 6381 df-ec 6383 df-qs 6387 df-ni 7054 df-pli 7055 df-mi 7056 df-lti 7057 df-plpq 7094 df-mpq 7095 df-enq 7097 df-nqqs 7098 df-plqqs 7099 df-mqqs 7100 df-1nqqs 7101 df-rq 7102 df-ltnqqs 7103 df-inp 7216 df-i1p 7217 df-enr 7463 df-nr 7464 df-0r 7468 df-r 7551 |
This theorem is referenced by: axcaucvglemcl 7624 axcaucvglemval 7626 |
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