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Theorem elreal2 11163
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 11156 . . 3 ℝ = (R × {0R})
21eleq2i 2821 . 2 (𝐴 ∈ ℝ ↔ 𝐴 ∈ (R × {0R}))
3 xp1st 8031 . . . 4 (𝐴 ∈ (R × {0R}) → (1st𝐴) ∈ R)
4 1st2nd2 8038 . . . . 5 (𝐴 ∈ (R × {0R}) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
5 xp2nd 8032 . . . . . . 7 (𝐴 ∈ (R × {0R}) → (2nd𝐴) ∈ {0R})
6 elsni 4649 . . . . . . 7 ((2nd𝐴) ∈ {0R} → (2nd𝐴) = 0R)
75, 6syl 17 . . . . . 6 (𝐴 ∈ (R × {0R}) → (2nd𝐴) = 0R)
87opeq2d 4885 . . . . 5 (𝐴 ∈ (R × {0R}) → ⟨(1st𝐴), (2nd𝐴)⟩ = ⟨(1st𝐴), 0R⟩)
94, 8eqtrd 2768 . . . 4 (𝐴 ∈ (R × {0R}) → 𝐴 = ⟨(1st𝐴), 0R⟩)
103, 9jca 510 . . 3 (𝐴 ∈ (R × {0R}) → ((1st𝐴) ∈ R𝐴 = ⟨(1st𝐴), 0R⟩))
11 eleq1 2817 . . . . 5 (𝐴 = ⟨(1st𝐴), 0R⟩ → (𝐴 ∈ (R × {0R}) ↔ ⟨(1st𝐴), 0R⟩ ∈ (R × {0R})))
12 0r 11111 . . . . . . . 8 0RR
1312elexi 3493 . . . . . . 7 0R ∈ V
1413snid 4669 . . . . . 6 0R ∈ {0R}
15 opelxp 5718 . . . . . 6 (⟨(1st𝐴), 0R⟩ ∈ (R × {0R}) ↔ ((1st𝐴) ∈ R ∧ 0R ∈ {0R}))
1614, 15mpbiran2 708 . . . . 5 (⟨(1st𝐴), 0R⟩ ∈ (R × {0R}) ↔ (1st𝐴) ∈ R)
1711, 16bitrdi 286 . . . 4 (𝐴 = ⟨(1st𝐴), 0R⟩ → (𝐴 ∈ (R × {0R}) ↔ (1st𝐴) ∈ R))
1817biimparc 478 . . 3 (((1st𝐴) ∈ R𝐴 = ⟨(1st𝐴), 0R⟩) → 𝐴 ∈ (R × {0R}))
1910, 18impbii 208 . 2 (𝐴 ∈ (R × {0R}) ↔ ((1st𝐴) ∈ R𝐴 = ⟨(1st𝐴), 0R⟩))
202, 19bitri 274 1 (𝐴 ∈ ℝ ↔ ((1st𝐴) ∈ R𝐴 = ⟨(1st𝐴), 0R⟩))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1533  wcel 2098  {csn 4632  cop 4638   × cxp 5680  cfv 6553  1st c1st 7997  2nd c2nd 7998  Rcnr 10896  0Rc0r 10897  cr 11145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9672
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-oadd 8497  df-omul 8498  df-er 8731  df-ec 8733  df-qs 8737  df-ni 10903  df-pli 10904  df-mi 10905  df-lti 10906  df-plpq 10939  df-mpq 10940  df-ltpq 10941  df-enq 10942  df-nq 10943  df-erq 10944  df-plq 10945  df-mq 10946  df-1nq 10947  df-rq 10948  df-ltnq 10949  df-np 11012  df-1p 11013  df-enr 11086  df-nr 11087  df-0r 11091  df-r 11156
This theorem is referenced by:  ltresr2  11172  axrnegex  11193  axpre-sup  11200
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