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Mirrors > Home > ILE Home > Th. List > elrealeu | GIF version |
Description: The real number mapping in elreal 7760 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 7760 | . . . 4 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | |
2 | 1 | biimpi 119 | . . 3 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) |
3 | eqtr3 2184 | . . . . . . . 8 ⊢ ((〈𝑥, 0R〉 = 𝐴 ∧ 〈𝑦, 0R〉 = 𝐴) → 〈𝑥, 0R〉 = 〈𝑦, 0R〉) | |
4 | 0r 7682 | . . . . . . . . . 10 ⊢ 0R ∈ R | |
5 | opthg 4210 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ R ∧ 0R ∈ R) → (〈𝑥, 0R〉 = 〈𝑦, 0R〉 ↔ (𝑥 = 𝑦 ∧ 0R = 0R))) | |
6 | 4, 5 | mpan2 422 | . . . . . . . . 9 ⊢ (𝑥 ∈ R → (〈𝑥, 0R〉 = 〈𝑦, 0R〉 ↔ (𝑥 = 𝑦 ∧ 0R = 0R))) |
7 | 6 | ad2antlr 481 | . . . . . . . 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 2537 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ R) → ∀𝑦 ∈ R ((〈𝑥, 0R〉 = 𝐴 ∧ 〈𝑦, 0R〉 = 𝐴) → 𝑥 = 𝑦)) |
12 | 11 | ralrimiva 2537 | . . . 4 ⊢ (𝐴 ∈ ℝ → ∀𝑥 ∈ R ∀𝑦 ∈ R ((〈𝑥, 0R〉 = 𝐴 ∧ 〈𝑦, 0R〉 = 𝐴) → 𝑥 = 𝑦)) |
13 | opeq1 3752 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 〈𝑥, 0R〉 = 〈𝑦, 0R〉) | |
14 | 13 | eqeq1d 2173 | . . . . 5 ⊢ (𝑥 = 𝑦 → (〈𝑥, 0R〉 = 𝐴 ↔ 〈𝑦, 0R〉 = 𝐴)) |
15 | 14 | rmo4 2914 | . . . 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 2676 | . . 3 ⊢ (∃!𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴 ↔ (∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴 ∧ ∃*𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴)) | |
18 | 2, 16, 17 | sylanbrc 414 | . 2 ⊢ (𝐴 ∈ ℝ → ∃!𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) |
19 | reurex 2677 | . . 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 1342 ∈ wcel 2135 ∀wral 2442 ∃wrex 2443 ∃!wreu 2444 ∃*wrmo 2445 〈cop 3573 Rcnr 7229 0Rc0r 7230 ℝcr 7743 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-eprel 4261 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-1o 6375 df-oadd 6379 df-omul 6380 df-er 6492 df-ec 6494 df-qs 6498 df-ni 7236 df-pli 7237 df-mi 7238 df-lti 7239 df-plpq 7276 df-mpq 7277 df-enq 7279 df-nqqs 7280 df-plqqs 7281 df-mqqs 7282 df-1nqqs 7283 df-rq 7284 df-ltnqqs 7285 df-inp 7398 df-i1p 7399 df-enr 7658 df-nr 7659 df-0r 7663 df-r 7754 |
This theorem is referenced by: axcaucvglemcl 7827 axcaucvglemval 7829 |
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