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Theorem elrealeu 7558
Description: The real number mapping in elreal 7557 is unique. (Contributed by Jim Kingdon, 11-Jul-2021.)
Assertion
Ref Expression
elrealeu (𝐴 ∈ ℝ ↔ ∃!𝑥R𝑥, 0R⟩ = 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem elrealeu
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elreal 7557 . . . 4 (𝐴 ∈ ℝ ↔ ∃𝑥R𝑥, 0R⟩ = 𝐴)
21biimpi 119 . . 3 (𝐴 ∈ ℝ → ∃𝑥R𝑥, 0R⟩ = 𝐴)
3 eqtr3 2132 . . . . . . . 8 ((⟨𝑥, 0R⟩ = 𝐴 ∧ ⟨𝑦, 0R⟩ = 𝐴) → ⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩)
4 0r 7487 . . . . . . . . . 10 0RR
5 opthg 4118 . . . . . . . . . 10 ((𝑥R ∧ 0RR) → (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ↔ (𝑥 = 𝑦 ∧ 0R = 0R)))
64, 5mpan2 419 . . . . . . . . 9 (𝑥R → (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ↔ (𝑥 = 𝑦 ∧ 0R = 0R)))
76ad2antlr 478 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ 𝑥R) ∧ 𝑦R) → (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ↔ (𝑥 = 𝑦 ∧ 0R = 0R)))
83, 7syl5ib 153 . . . . . . 7 (((𝐴 ∈ ℝ ∧ 𝑥R) ∧ 𝑦R) → ((⟨𝑥, 0R⟩ = 𝐴 ∧ ⟨𝑦, 0R⟩ = 𝐴) → (𝑥 = 𝑦 ∧ 0R = 0R)))
9 simpl 108 . . . . . . 7 ((𝑥 = 𝑦 ∧ 0R = 0R) → 𝑥 = 𝑦)
108, 9syl6 33 . . . . . 6 (((𝐴 ∈ ℝ ∧ 𝑥R) ∧ 𝑦R) → ((⟨𝑥, 0R⟩ = 𝐴 ∧ ⟨𝑦, 0R⟩ = 𝐴) → 𝑥 = 𝑦))
1110ralrimiva 2477 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝑥R) → ∀𝑦R ((⟨𝑥, 0R⟩ = 𝐴 ∧ ⟨𝑦, 0R⟩ = 𝐴) → 𝑥 = 𝑦))
1211ralrimiva 2477 . . . 4 (𝐴 ∈ ℝ → ∀𝑥R𝑦R ((⟨𝑥, 0R⟩ = 𝐴 ∧ ⟨𝑦, 0R⟩ = 𝐴) → 𝑥 = 𝑦))
13 opeq1 3669 . . . . . 6 (𝑥 = 𝑦 → ⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩)
1413eqeq1d 2121 . . . . 5 (𝑥 = 𝑦 → (⟨𝑥, 0R⟩ = 𝐴 ↔ ⟨𝑦, 0R⟩ = 𝐴))
1514rmo4 2844 . . . 4 (∃*𝑥R𝑥, 0R⟩ = 𝐴 ↔ ∀𝑥R𝑦R ((⟨𝑥, 0R⟩ = 𝐴 ∧ ⟨𝑦, 0R⟩ = 𝐴) → 𝑥 = 𝑦))
1612, 15sylibr 133 . . 3 (𝐴 ∈ ℝ → ∃*𝑥R𝑥, 0R⟩ = 𝐴)
17 reu5 2615 . . 3 (∃!𝑥R𝑥, 0R⟩ = 𝐴 ↔ (∃𝑥R𝑥, 0R⟩ = 𝐴 ∧ ∃*𝑥R𝑥, 0R⟩ = 𝐴))
182, 16, 17sylanbrc 411 . 2 (𝐴 ∈ ℝ → ∃!𝑥R𝑥, 0R⟩ = 𝐴)
19 reurex 2616 . . 3 (∃!𝑥R𝑥, 0R⟩ = 𝐴 → ∃𝑥R𝑥, 0R⟩ = 𝐴)
2019, 1sylibr 133 . 2 (∃!𝑥R𝑥, 0R⟩ = 𝐴𝐴 ∈ ℝ)
2118, 20impbii 125 1 (𝐴 ∈ ℝ ↔ ∃!𝑥R𝑥, 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  cr 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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