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Theorem elreal2 7368
Description: Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.)
Assertion
Ref Expression
elreal2 (𝐴 ∈ ℝ ↔ ((1st𝐴) ∈ R𝐴 = ⟨(1st𝐴), 0R⟩))

Proof of Theorem elreal2
StepHypRef Expression
1 df-r 7360 . . 3 ℝ = (R × {0R})
21eleq2i 2154 . 2 (𝐴 ∈ ℝ ↔ 𝐴 ∈ (R × {0R}))
3 xp1st 5936 . . . 4 (𝐴 ∈ (R × {0R}) → (1st𝐴) ∈ R)
4 1st2nd2 5945 . . . . 5 (𝐴 ∈ (R × {0R}) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
5 xp2nd 5937 . . . . . . 7 (𝐴 ∈ (R × {0R}) → (2nd𝐴) ∈ {0R})
6 elsni 3464 . . . . . . 7 ((2nd𝐴) ∈ {0R} → (2nd𝐴) = 0R)
75, 6syl 14 . . . . . 6 (𝐴 ∈ (R × {0R}) → (2nd𝐴) = 0R)
87opeq2d 3629 . . . . 5 (𝐴 ∈ (R × {0R}) → ⟨(1st𝐴), (2nd𝐴)⟩ = ⟨(1st𝐴), 0R⟩)
94, 8eqtrd 2120 . . . 4 (𝐴 ∈ (R × {0R}) → 𝐴 = ⟨(1st𝐴), 0R⟩)
103, 9jca 300 . . 3 (𝐴 ∈ (R × {0R}) → ((1st𝐴) ∈ R𝐴 = ⟨(1st𝐴), 0R⟩))
11 eleq1 2150 . . . . 5 (𝐴 = ⟨(1st𝐴), 0R⟩ → (𝐴 ∈ (R × {0R}) ↔ ⟨(1st𝐴), 0R⟩ ∈ (R × {0R})))
12 0r 7296 . . . . . . . 8 0RR
1312elexi 2631 . . . . . . 7 0R ∈ V
1413snid 3475 . . . . . 6 0R ∈ {0R}
15 opelxp 4467 . . . . . 6 (⟨(1st𝐴), 0R⟩ ∈ (R × {0R}) ↔ ((1st𝐴) ∈ R ∧ 0R ∈ {0R}))
1614, 15mpbiran2 887 . . . . 5 (⟨(1st𝐴), 0R⟩ ∈ (R × {0R}) ↔ (1st𝐴) ∈ R)
1711, 16syl6bb 194 . . . 4 (𝐴 = ⟨(1st𝐴), 0R⟩ → (𝐴 ∈ (R × {0R}) ↔ (1st𝐴) ∈ R))
1817biimparc 293 . . 3 (((1st𝐴) ∈ R𝐴 = ⟨(1st𝐴), 0R⟩) → 𝐴 ∈ (R × {0R}))
1910, 18impbii 124 . 2 (𝐴 ∈ (R × {0R}) ↔ ((1st𝐴) ∈ R𝐴 = ⟨(1st𝐴), 0R⟩))
202, 19bitri 182 1 (𝐴 ∈ ℝ ↔ ((1st𝐴) ∈ R𝐴 = ⟨(1st𝐴), 0R⟩))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1289  wcel 1438  {csn 3446  cop 3449   × cxp 4436  cfv 5015  1st c1st 5909  2nd c2nd 5910  Rcnr 6856  0Rc0r 6857  cr 7349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-1o 6181  df-oadd 6185  df-omul 6186  df-er 6292  df-ec 6294  df-qs 6298  df-ni 6863  df-pli 6864  df-mi 6865  df-lti 6866  df-plpq 6903  df-mpq 6904  df-enq 6906  df-nqqs 6907  df-plqqs 6908  df-mqqs 6909  df-1nqqs 6910  df-rq 6911  df-ltnqqs 6912  df-inp 7025  df-i1p 7026  df-enr 7272  df-nr 7273  df-0r 7277  df-r 7360
This theorem is referenced by:  ltresr2  7377  axrnegex  7414
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