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Theorem elreal2 10541
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 10534 . . 3 ℝ = (R × {0R})
21eleq2i 2907 . 2 (𝐴 ∈ ℝ ↔ 𝐴 ∈ (R × {0R}))
3 xp1st 7706 . . . 4 (𝐴 ∈ (R × {0R}) → (1st𝐴) ∈ R)
4 1st2nd2 7713 . . . . 5 (𝐴 ∈ (R × {0R}) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
5 xp2nd 7707 . . . . . . 7 (𝐴 ∈ (R × {0R}) → (2nd𝐴) ∈ {0R})
6 elsni 4565 . . . . . . 7 ((2nd𝐴) ∈ {0R} → (2nd𝐴) = 0R)
75, 6syl 17 . . . . . 6 (𝐴 ∈ (R × {0R}) → (2nd𝐴) = 0R)
87opeq2d 4793 . . . . 5 (𝐴 ∈ (R × {0R}) → ⟨(1st𝐴), (2nd𝐴)⟩ = ⟨(1st𝐴), 0R⟩)
94, 8eqtrd 2859 . . . 4 (𝐴 ∈ (R × {0R}) → 𝐴 = ⟨(1st𝐴), 0R⟩)
103, 9jca 515 . . 3 (𝐴 ∈ (R × {0R}) → ((1st𝐴) ∈ R𝐴 = ⟨(1st𝐴), 0R⟩))
11 eleq1 2903 . . . . 5 (𝐴 = ⟨(1st𝐴), 0R⟩ → (𝐴 ∈ (R × {0R}) ↔ ⟨(1st𝐴), 0R⟩ ∈ (R × {0R})))
12 0r 10489 . . . . . . . 8 0RR
1312elexi 3498 . . . . . . 7 0R ∈ V
1413snid 4584 . . . . . 6 0R ∈ {0R}
15 opelxp 5574 . . . . . 6 (⟨(1st𝐴), 0R⟩ ∈ (R × {0R}) ↔ ((1st𝐴) ∈ R ∧ 0R ∈ {0R}))
1614, 15mpbiran2 709 . . . . 5 (⟨(1st𝐴), 0R⟩ ∈ (R × {0R}) ↔ (1st𝐴) ∈ R)
1711, 16syl6bb 290 . . . 4 (𝐴 = ⟨(1st𝐴), 0R⟩ → (𝐴 ∈ (R × {0R}) ↔ (1st𝐴) ∈ R))
1817biimparc 483 . . 3 (((1st𝐴) ∈ R𝐴 = ⟨(1st𝐴), 0R⟩) → 𝐴 ∈ (R × {0R}))
1910, 18impbii 212 . 2 (𝐴 ∈ (R × {0R}) ↔ ((1st𝐴) ∈ R𝐴 = ⟨(1st𝐴), 0R⟩))
202, 19bitri 278 1 (𝐴 ∈ ℝ ↔ ((1st𝐴) ∈ R𝐴 = ⟨(1st𝐴), 0R⟩))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2115  {csn 4548  cop 4554   × cxp 5536  cfv 6338  1st c1st 7672  2nd c2nd 7673  Rcnr 10274  0Rc0r 10275  cr 10523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313  ax-un 7446  ax-inf2 9090
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rmo 3140  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4822  df-int 4860  df-iun 4904  df-br 5050  df-opab 5112  df-mpt 5130  df-tr 5156  df-id 5443  df-eprel 5448  df-po 5457  df-so 5458  df-fr 5497  df-we 5499  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6297  df-fun 6340  df-fn 6341  df-f 6342  df-f1 6343  df-fo 6344  df-f1o 6345  df-fv 6346  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7674  df-2nd 7675  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-1o 8087  df-oadd 8091  df-omul 8092  df-er 8274  df-ec 8276  df-qs 8280  df-ni 10281  df-pli 10282  df-mi 10283  df-lti 10284  df-plpq 10317  df-mpq 10318  df-ltpq 10319  df-enq 10320  df-nq 10321  df-erq 10322  df-plq 10323  df-mq 10324  df-1nq 10325  df-rq 10326  df-ltnq 10327  df-np 10390  df-1p 10391  df-enr 10464  df-nr 10465  df-0r 10469  df-r 10534
This theorem is referenced by:  ltresr2  10550  axrnegex  10571  axpre-sup  10578
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