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Theorem elrealeu 7841
Description: The real number mapping in elreal 7840 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 7840 . . . 4 (𝐴 ∈ ℝ ↔ ∃𝑥R𝑥, 0R⟩ = 𝐴)
21biimpi 120 . . 3 (𝐴 ∈ ℝ → ∃𝑥R𝑥, 0R⟩ = 𝐴)
3 eqtr3 2207 . . . . . . . 8 ((⟨𝑥, 0R⟩ = 𝐴 ∧ ⟨𝑦, 0R⟩ = 𝐴) → ⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩)
4 0r 7762 . . . . . . . . . 10 0RR
5 opthg 4250 . . . . . . . . . 10 ((𝑥R ∧ 0RR) → (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ↔ (𝑥 = 𝑦 ∧ 0R = 0R)))
64, 5mpan2 425 . . . . . . . . 9 (𝑥R → (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ↔ (𝑥 = 𝑦 ∧ 0R = 0R)))
76ad2antlr 489 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ 𝑥R) ∧ 𝑦R) → (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ↔ (𝑥 = 𝑦 ∧ 0R = 0R)))
83, 7imbitrid 154 . . . . . . 7 (((𝐴 ∈ ℝ ∧ 𝑥R) ∧ 𝑦R) → ((⟨𝑥, 0R⟩ = 𝐴 ∧ ⟨𝑦, 0R⟩ = 𝐴) → (𝑥 = 𝑦 ∧ 0R = 0R)))
9 simpl 109 . . . . . . 7 ((𝑥 = 𝑦 ∧ 0R = 0R) → 𝑥 = 𝑦)
108, 9syl6 33 . . . . . 6 (((𝐴 ∈ ℝ ∧ 𝑥R) ∧ 𝑦R) → ((⟨𝑥, 0R⟩ = 𝐴 ∧ ⟨𝑦, 0R⟩ = 𝐴) → 𝑥 = 𝑦))
1110ralrimiva 2560 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝑥R) → ∀𝑦R ((⟨𝑥, 0R⟩ = 𝐴 ∧ ⟨𝑦, 0R⟩ = 𝐴) → 𝑥 = 𝑦))
1211ralrimiva 2560 . . . 4 (𝐴 ∈ ℝ → ∀𝑥R𝑦R ((⟨𝑥, 0R⟩ = 𝐴 ∧ ⟨𝑦, 0R⟩ = 𝐴) → 𝑥 = 𝑦))
13 opeq1 3790 . . . . . 6 (𝑥 = 𝑦 → ⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩)
1413eqeq1d 2196 . . . . 5 (𝑥 = 𝑦 → (⟨𝑥, 0R⟩ = 𝐴 ↔ ⟨𝑦, 0R⟩ = 𝐴))
1514rmo4 2942 . . . 4 (∃*𝑥R𝑥, 0R⟩ = 𝐴 ↔ ∀𝑥R𝑦R ((⟨𝑥, 0R⟩ = 𝐴 ∧ ⟨𝑦, 0R⟩ = 𝐴) → 𝑥 = 𝑦))
1612, 15sylibr 134 . . 3 (𝐴 ∈ ℝ → ∃*𝑥R𝑥, 0R⟩ = 𝐴)
17 reu5 2700 . . 3 (∃!𝑥R𝑥, 0R⟩ = 𝐴 ↔ (∃𝑥R𝑥, 0R⟩ = 𝐴 ∧ ∃*𝑥R𝑥, 0R⟩ = 𝐴))
182, 16, 17sylanbrc 417 . 2 (𝐴 ∈ ℝ → ∃!𝑥R𝑥, 0R⟩ = 𝐴)
19 reurex 2701 . . 3 (∃!𝑥R𝑥, 0R⟩ = 𝐴 → ∃𝑥R𝑥, 0R⟩ = 𝐴)
2019, 1sylibr 134 . 2 (∃!𝑥R𝑥, 0R⟩ = 𝐴𝐴 ∈ ℝ)
2118, 20impbii 126 1 (𝐴 ∈ ℝ ↔ ∃!𝑥R𝑥, 0R⟩ = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1363  wcel 2158  wral 2465  wrex 2466  ∃!wreu 2467  ∃*wrmo 2468  cop 3607  Rcnr 7309  0Rc0r 7310  cr 7823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-eprel 4301  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-irdg 6384  df-1o 6430  df-oadd 6434  df-omul 6435  df-er 6548  df-ec 6550  df-qs 6554  df-ni 7316  df-pli 7317  df-mi 7318  df-lti 7319  df-plpq 7356  df-mpq 7357  df-enq 7359  df-nqqs 7360  df-plqqs 7361  df-mqqs 7362  df-1nqqs 7363  df-rq 7364  df-ltnqqs 7365  df-inp 7478  df-i1p 7479  df-enr 7738  df-nr 7739  df-0r 7743  df-r 7834
This theorem is referenced by:  axcaucvglemcl  7907  axcaucvglemval  7909
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