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Mirrors > Home > ILE Home > Th. List > elreal2 | GIF version |
Description: Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.) |
Ref | Expression |
---|---|
elreal2 | ⊢ (A ∈ ℝ ↔ ((1st ‘A) ∈ R ∧ A = 〈(1st ‘A), 0R〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-r 6721 | . . 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 6678 | . . . . . . . 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〉)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 = wceq 1242 ∈ wcel 1390 {csn 3367 〈cop 3370 × cxp 4286 ‘cfv 4845 1st c1st 5707 2nd c2nd 5708 Rcnr 6281 0Rc0r 6282 ℝcr 6710 |
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-bndl 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 6449 df-i1p 6450 df-enr 6654 df-nr 6655 df-0r 6659 df-r 6721 |
This theorem is referenced by: ltresr2 6737 axrnegex 6763 |
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