Theorem elreal2 6688

Description: Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.)

Assertion

Ref	Expression
elreal2	⊢ (A ∈ ℝ ↔ ((1st ‘A) ∈ R ∧ A = ⟨(1st ‘A), 0R⟩))

Proof of Theorem elreal2

Step	Hyp	Ref	Expression
1		df-r 6681	. . 3 ⊢ ℝ = (R × {0R})
2	1	eleq2i 2101	. 2 ⊢ (A ∈ ℝ ↔ A ∈ (R × {0R}))
3		xp1st 5734	. . . 4 ⊢ (A ∈ (R × {0R}) → (1st ‘A) ∈ R)
4		1st2nd2 5743	. . . . 5 ⊢ (A ∈ (R × {0R}) → A = ⟨(1st ‘A), (2nd ‘A)⟩)
5		xp2nd 5735	. . . . . . 7 ⊢ (A ∈ (R × {0R}) → (2nd ‘A) ∈ {0R})
6		elsni 3391	. . . . . . 7 ⊢ ((2nd ‘A) ∈ {0R} → (2nd ‘A) = 0R)
7	5, 6	syl 14	. . . . . 6 ⊢ (A ∈ (R × {0R}) → (2nd ‘A) = 0R)
8	7	opeq2d 3547	. . . . 5 ⊢ (A ∈ (R × {0R}) → ⟨(1st ‘A), (2nd ‘A)⟩ = ⟨(1st ‘A), 0R⟩)
9	4, 8	eqtrd 2069	. . . 4 ⊢ (A ∈ (R × {0R}) → A = ⟨(1st ‘A), 0R⟩)
10	3, 9	jca 290	. . 3 ⊢ (A ∈ (R × {0R}) → ((1st ‘A) ∈ R ∧ A = ⟨(1st ‘A), 0R⟩))
11		eleq1 2097	. . . . 5 ⊢ (A = ⟨(1st ‘A), 0R⟩ → (A ∈ (R × {0R}) ↔ ⟨(1st ‘A), 0R⟩ ∈ (R × {0R})))
12		0r 6638	. . . . . . . 8 ⊢ 0R ∈ R
13	12	elexi 2561	. . . . . . 7 ⊢ 0R ∈ V
14	13	snid 3394	. . . . . 6 ⊢ 0R ∈ {0R}
15		opelxp 4317	. . . . . 6 ⊢ (⟨(1st ‘A), 0R⟩ ∈ (R × {0R}) ↔ ((1st ‘A) ∈ R ∧ 0R ∈ {0R}))
16	14, 15	mpbiran2 847	. . . . 5 ⊢ (⟨(1st ‘A), 0R⟩ ∈ (R × {0R}) ↔ (1st ‘A) ∈ R)
17	11, 16	syl6bb 185	. . . 4 ⊢ (A = ⟨(1st ‘A), 0R⟩ → (A ∈ (R × {0R}) ↔ (1st ‘A) ∈ R))
18	17	biimparc 283	. . 3 ⊢ (((1st ‘A) ∈ R ∧ A = ⟨(1st ‘A), 0R⟩) → A ∈ (R × {0R}))
19	10, 18	impbii 117	. 2 ⊢ (A ∈ (R × {0R}) ↔ ((1st ‘A) ∈ R ∧ A = ⟨(1st ‘A), 0R⟩))
20	2, 19	bitri 173	1 ⊢ (A ∈ ℝ ↔ ((1st ‘A) ∈ R ∧ A = ⟨(1st ‘A), 0R⟩))

Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  {csn 3367  ⟨cop 3370   × cxp 4286  ‘cfv 4845  1^st c1st 5707  2^nd c2nd 5708  Rcnr 6281  0_Rc0r 6282  ℝcr 6670

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-inp 6448  df-i1p 6449  df-enr 6614  df-nr 6615  df-0r 6619  df-r 6681

This theorem is referenced by:  ltresr2  6697  axrnegex  6723