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Theorem elreal2 9991
Description: Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
Assertion
Ref Expression
elreal2 (𝐴 ∈ ℝ ↔ ((1st𝐴) ∈ R𝐴 = ⟨(1st𝐴), 0R⟩))

Proof of Theorem elreal2
StepHypRef Expression
1 df-r 9984 . . 3 ℝ = (R × {0R})
21eleq2i 2722 . 2 (𝐴 ∈ ℝ ↔ 𝐴 ∈ (R × {0R}))
3 xp1st 7242 . . . 4 (𝐴 ∈ (R × {0R}) → (1st𝐴) ∈ R)
4 1st2nd2 7249 . . . . 5 (𝐴 ∈ (R × {0R}) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
5 xp2nd 7243 . . . . . . 7 (𝐴 ∈ (R × {0R}) → (2nd𝐴) ∈ {0R})
6 elsni 4227 . . . . . . 7 ((2nd𝐴) ∈ {0R} → (2nd𝐴) = 0R)
75, 6syl 17 . . . . . 6 (𝐴 ∈ (R × {0R}) → (2nd𝐴) = 0R)
87opeq2d 4440 . . . . 5 (𝐴 ∈ (R × {0R}) → ⟨(1st𝐴), (2nd𝐴)⟩ = ⟨(1st𝐴), 0R⟩)
94, 8eqtrd 2685 . . . 4 (𝐴 ∈ (R × {0R}) → 𝐴 = ⟨(1st𝐴), 0R⟩)
103, 9jca 553 . . 3 (𝐴 ∈ (R × {0R}) → ((1st𝐴) ∈ R𝐴 = ⟨(1st𝐴), 0R⟩))
11 eleq1 2718 . . . . 5 (𝐴 = ⟨(1st𝐴), 0R⟩ → (𝐴 ∈ (R × {0R}) ↔ ⟨(1st𝐴), 0R⟩ ∈ (R × {0R})))
12 0r 9939 . . . . . . . 8 0RR
1312elexi 3244 . . . . . . 7 0R ∈ V
1413snid 4241 . . . . . 6 0R ∈ {0R}
15 opelxp 5180 . . . . . 6 (⟨(1st𝐴), 0R⟩ ∈ (R × {0R}) ↔ ((1st𝐴) ∈ R ∧ 0R ∈ {0R}))
1614, 15mpbiran2 974 . . . . 5 (⟨(1st𝐴), 0R⟩ ∈ (R × {0R}) ↔ (1st𝐴) ∈ R)
1711, 16syl6bb 276 . . . 4 (𝐴 = ⟨(1st𝐴), 0R⟩ → (𝐴 ∈ (R × {0R}) ↔ (1st𝐴) ∈ R))
1817biimparc 503 . . 3 (((1st𝐴) ∈ R𝐴 = ⟨(1st𝐴), 0R⟩) → 𝐴 ∈ (R × {0R}))
1910, 18impbii 199 . 2 (𝐴 ∈ (R × {0R}) ↔ ((1st𝐴) ∈ R𝐴 = ⟨(1st𝐴), 0R⟩))
202, 19bitri 264 1 (𝐴 ∈ ℝ ↔ ((1st𝐴) ∈ R𝐴 = ⟨(1st𝐴), 0R⟩))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wcel 2030  {csn 4210  cop 4216   × cxp 5141  cfv 5926  1st c1st 7208  2nd c2nd 7209  Rcnr 9725  0Rc0r 9726  cr 9973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-ec 7789  df-qs 7793  df-ni 9732  df-pli 9733  df-mi 9734  df-lti 9735  df-plpq 9768  df-mpq 9769  df-ltpq 9770  df-enq 9771  df-nq 9772  df-erq 9773  df-plq 9774  df-mq 9775  df-1nq 9776  df-rq 9777  df-ltnq 9778  df-np 9841  df-1p 9842  df-enr 9915  df-nr 9916  df-0r 9920  df-r 9984
This theorem is referenced by:  ltresr2  10000  axrnegex  10021  axpre-sup  10028
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