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Theorem elrealeu 7656
Description: The real number mapping in elreal 7655 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 7655 . . . 4 (𝐴 ∈ ℝ ↔ ∃𝑥R𝑥, 0R⟩ = 𝐴)
21biimpi 119 . . 3 (𝐴 ∈ ℝ → ∃𝑥R𝑥, 0R⟩ = 𝐴)
3 eqtr3 2159 . . . . . . . 8 ((⟨𝑥, 0R⟩ = 𝐴 ∧ ⟨𝑦, 0R⟩ = 𝐴) → ⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩)
4 0r 7577 . . . . . . . . . 10 0RR
5 opthg 4163 . . . . . . . . . 10 ((𝑥R ∧ 0RR) → (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ↔ (𝑥 = 𝑦 ∧ 0R = 0R)))
64, 5mpan2 421 . . . . . . . . 9 (𝑥R → (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ↔ (𝑥 = 𝑦 ∧ 0R = 0R)))
76ad2antlr 480 . . . . . . . 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 2505 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝑥R) → ∀𝑦R ((⟨𝑥, 0R⟩ = 𝐴 ∧ ⟨𝑦, 0R⟩ = 𝐴) → 𝑥 = 𝑦))
1211ralrimiva 2505 . . . 4 (𝐴 ∈ ℝ → ∀𝑥R𝑦R ((⟨𝑥, 0R⟩ = 𝐴 ∧ ⟨𝑦, 0R⟩ = 𝐴) → 𝑥 = 𝑦))
13 opeq1 3708 . . . . . 6 (𝑥 = 𝑦 → ⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩)
1413eqeq1d 2148 . . . . 5 (𝑥 = 𝑦 → (⟨𝑥, 0R⟩ = 𝐴 ↔ ⟨𝑦, 0R⟩ = 𝐴))
1514rmo4 2877 . . . 4 (∃*𝑥R𝑥, 0R⟩ = 𝐴 ↔ ∀𝑥R𝑦R ((⟨𝑥, 0R⟩ = 𝐴 ∧ ⟨𝑦, 0R⟩ = 𝐴) → 𝑥 = 𝑦))
1612, 15sylibr 133 . . 3 (𝐴 ∈ ℝ → ∃*𝑥R𝑥, 0R⟩ = 𝐴)
17 reu5 2643 . . 3 (∃!𝑥R𝑥, 0R⟩ = 𝐴 ↔ (∃𝑥R𝑥, 0R⟩ = 𝐴 ∧ ∃*𝑥R𝑥, 0R⟩ = 𝐴))
182, 16, 17sylanbrc 413 . 2 (𝐴 ∈ ℝ → ∃!𝑥R𝑥, 0R⟩ = 𝐴)
19 reurex 2644 . . 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 1331  wcel 1480  wral 2416  wrex 2417  ∃!wreu 2418  ∃*wrmo 2419  cop 3530  Rcnr 7124  0Rc0r 7125  cr 7638
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4046  ax-sep 4049  ax-nul 4057  ax-pow 4101  ax-pr 4134  ax-un 4358  ax-setind 4455  ax-iinf 4505
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-int 3775  df-iun 3818  df-br 3933  df-opab 3993  df-mpt 3994  df-tr 4030  df-eprel 4214  df-id 4218  df-po 4221  df-iso 4222  df-iord 4291  df-on 4293  df-suc 4296  df-iom 4508  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-rn 4553  df-res 4554  df-ima 4555  df-iota 5091  df-fun 5128  df-fn 5129  df-f 5130  df-f1 5131  df-fo 5132  df-f1o 5133  df-fv 5134  df-ov 5780  df-oprab 5781  df-mpo 5782  df-1st 6041  df-2nd 6042  df-recs 6205  df-irdg 6270  df-1o 6316  df-oadd 6320  df-omul 6321  df-er 6432  df-ec 6434  df-qs 6438  df-ni 7131  df-pli 7132  df-mi 7133  df-lti 7134  df-plpq 7171  df-mpq 7172  df-enq 7174  df-nqqs 7175  df-plqqs 7176  df-mqqs 7177  df-1nqqs 7178  df-rq 7179  df-ltnqqs 7180  df-inp 7293  df-i1p 7294  df-enr 7553  df-nr 7554  df-0r 7558  df-r 7649
This theorem is referenced by:  axcaucvglemcl  7722  axcaucvglemval  7724
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