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Mirrors > Home > ILE Home > Th. List > elrealeu | GIF version |
Description: The real number mapping in elreal 6905 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 6905 | . . . 4 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | |
2 | 1 | biimpi 113 | . . 3 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) |
3 | eqtr3 2059 | . . . . . . . 8 ⊢ ((〈𝑥, 0R〉 = 𝐴 ∧ 〈𝑦, 0R〉 = 𝐴) → 〈𝑥, 0R〉 = 〈𝑦, 0R〉) | |
4 | 0r 6835 | . . . . . . . . . 10 ⊢ 0R ∈ R | |
5 | opthg 3975 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ R ∧ 0R ∈ R) → (〈𝑥, 0R〉 = 〈𝑦, 0R〉 ↔ (𝑥 = 𝑦 ∧ 0R = 0R))) | |
6 | 4, 5 | mpan2 401 | . . . . . . . . 9 ⊢ (𝑥 ∈ R → (〈𝑥, 0R〉 = 〈𝑦, 0R〉 ↔ (𝑥 = 𝑦 ∧ 0R = 0R))) |
7 | 6 | ad2antlr 458 | . . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ R) ∧ 𝑦 ∈ R) → (〈𝑥, 0R〉 = 〈𝑦, 0R〉 ↔ (𝑥 = 𝑦 ∧ 0R = 0R))) |
8 | 3, 7 | syl5ib 143 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ R) ∧ 𝑦 ∈ R) → ((〈𝑥, 0R〉 = 𝐴 ∧ 〈𝑦, 0R〉 = 𝐴) → (𝑥 = 𝑦 ∧ 0R = 0R))) |
9 | simpl 102 | . . . . . . 7 ⊢ ((𝑥 = 𝑦 ∧ 0R = 0R) → 𝑥 = 𝑦) | |
10 | 8, 9 | syl6 29 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ R) ∧ 𝑦 ∈ R) → ((〈𝑥, 0R〉 = 𝐴 ∧ 〈𝑦, 0R〉 = 𝐴) → 𝑥 = 𝑦)) |
11 | 10 | ralrimiva 2392 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ R) → ∀𝑦 ∈ R ((〈𝑥, 0R〉 = 𝐴 ∧ 〈𝑦, 0R〉 = 𝐴) → 𝑥 = 𝑦)) |
12 | 11 | ralrimiva 2392 | . . . 4 ⊢ (𝐴 ∈ ℝ → ∀𝑥 ∈ R ∀𝑦 ∈ R ((〈𝑥, 0R〉 = 𝐴 ∧ 〈𝑦, 0R〉 = 𝐴) → 𝑥 = 𝑦)) |
13 | opeq1 3549 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 〈𝑥, 0R〉 = 〈𝑦, 0R〉) | |
14 | 13 | eqeq1d 2048 | . . . . 5 ⊢ (𝑥 = 𝑦 → (〈𝑥, 0R〉 = 𝐴 ↔ 〈𝑦, 0R〉 = 𝐴)) |
15 | 14 | rmo4 2734 | . . . 4 ⊢ (∃*𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴 ↔ ∀𝑥 ∈ R ∀𝑦 ∈ R ((〈𝑥, 0R〉 = 𝐴 ∧ 〈𝑦, 0R〉 = 𝐴) → 𝑥 = 𝑦)) |
16 | 12, 15 | sylibr 137 | . . 3 ⊢ (𝐴 ∈ ℝ → ∃*𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) |
17 | reu5 2522 | . . 3 ⊢ (∃!𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴 ↔ (∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴 ∧ ∃*𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴)) | |
18 | 2, 16, 17 | sylanbrc 394 | . 2 ⊢ (𝐴 ∈ ℝ → ∃!𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) |
19 | reurex 2523 | . . 3 ⊢ (∃!𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴 → ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | |
20 | 19, 1 | sylibr 137 | . 2 ⊢ (∃!𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴 → 𝐴 ∈ ℝ) |
21 | 18, 20 | impbii 117 | 1 ⊢ (𝐴 ∈ ℝ ↔ ∃!𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 ∈ wcel 1393 ∀wral 2306 ∃wrex 2307 ∃!wreu 2308 ∃*wrmo 2309 〈cop 3378 Rcnr 6395 0Rc0r 6396 ℝcr 6888 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rmo 2314 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-inp 6564 df-i1p 6565 df-enr 6811 df-nr 6812 df-0r 6816 df-r 6899 |
This theorem is referenced by: axcaucvglemcl 6969 axcaucvglemval 6971 |
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