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Theorem elreal2 9900
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 9893 . . 3 ℝ = (R × {0R})
21eleq2i 2690 . 2 (𝐴 ∈ ℝ ↔ 𝐴 ∈ (R × {0R}))
3 xp1st 7146 . . . 4 (𝐴 ∈ (R × {0R}) → (1st𝐴) ∈ R)
4 1st2nd2 7153 . . . . 5 (𝐴 ∈ (R × {0R}) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
5 xp2nd 7147 . . . . . . 7 (𝐴 ∈ (R × {0R}) → (2nd𝐴) ∈ {0R})
6 elsni 4167 . . . . . . 7 ((2nd𝐴) ∈ {0R} → (2nd𝐴) = 0R)
75, 6syl 17 . . . . . 6 (𝐴 ∈ (R × {0R}) → (2nd𝐴) = 0R)
87opeq2d 4379 . . . . 5 (𝐴 ∈ (R × {0R}) → ⟨(1st𝐴), (2nd𝐴)⟩ = ⟨(1st𝐴), 0R⟩)
94, 8eqtrd 2655 . . . 4 (𝐴 ∈ (R × {0R}) → 𝐴 = ⟨(1st𝐴), 0R⟩)
103, 9jca 554 . . 3 (𝐴 ∈ (R × {0R}) → ((1st𝐴) ∈ R𝐴 = ⟨(1st𝐴), 0R⟩))
11 eleq1 2686 . . . . 5 (𝐴 = ⟨(1st𝐴), 0R⟩ → (𝐴 ∈ (R × {0R}) ↔ ⟨(1st𝐴), 0R⟩ ∈ (R × {0R})))
12 0r 9848 . . . . . . . 8 0RR
1312elexi 3199 . . . . . . 7 0R ∈ V
1413snid 4181 . . . . . 6 0R ∈ {0R}
15 opelxp 5108 . . . . . 6 (⟨(1st𝐴), 0R⟩ ∈ (R × {0R}) ↔ ((1st𝐴) ∈ R ∧ 0R ∈ {0R}))
1614, 15mpbiran2 953 . . . . 5 (⟨(1st𝐴), 0R⟩ ∈ (R × {0R}) ↔ (1st𝐴) ∈ R)
1711, 16syl6bb 276 . . . 4 (𝐴 = ⟨(1st𝐴), 0R⟩ → (𝐴 ∈ (R × {0R}) ↔ (1st𝐴) ∈ R))
1817biimparc 504 . . 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 384   = wceq 1480  wcel 1987  {csn 4150  cop 4156   × cxp 5074  cfv 5849  1st c1st 7114  2nd c2nd 7115  Rcnr 9634  0Rc0r 9635  cr 9882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-inf2 8485
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-oadd 7512  df-omul 7513  df-er 7690  df-ec 7692  df-qs 7696  df-ni 9641  df-pli 9642  df-mi 9643  df-lti 9644  df-plpq 9677  df-mpq 9678  df-ltpq 9679  df-enq 9680  df-nq 9681  df-erq 9682  df-plq 9683  df-mq 9684  df-1nq 9685  df-rq 9686  df-ltnq 9687  df-np 9750  df-1p 9751  df-enr 9824  df-nr 9825  df-0r 9829  df-r 9893
This theorem is referenced by:  ltresr2  9909  axrnegex  9930  axpre-sup  9937
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