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Mirrors > Home > ILE Home > Th. List > elreal | GIF version |
Description: Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.) |
Ref | Expression |
---|---|
elreal | ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-r 7421 | . . 3 ⊢ ℝ = (R × {0R}) | |
2 | 1 | eleq2i 2155 | . 2 ⊢ (𝐴 ∈ ℝ ↔ 𝐴 ∈ (R × {0R})) |
3 | elxp2 4470 | . . 3 ⊢ (𝐴 ∈ (R × {0R}) ↔ ∃𝑥 ∈ R ∃𝑦 ∈ {0R}𝐴 = 〈𝑥, 𝑦〉) | |
4 | 0r 7357 | . . . . . . 7 ⊢ 0R ∈ R | |
5 | 4 | elexi 2632 | . . . . . 6 ⊢ 0R ∈ V |
6 | opeq2 3629 | . . . . . . 7 ⊢ (𝑦 = 0R → 〈𝑥, 𝑦〉 = 〈𝑥, 0R〉) | |
7 | 6 | eqeq2d 2100 | . . . . . 6 ⊢ (𝑦 = 0R → (𝐴 = 〈𝑥, 𝑦〉 ↔ 𝐴 = 〈𝑥, 0R〉)) |
8 | 5, 7 | rexsn 3491 | . . . . 5 ⊢ (∃𝑦 ∈ {0R}𝐴 = 〈𝑥, 𝑦〉 ↔ 𝐴 = 〈𝑥, 0R〉) |
9 | eqcom 2091 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 0R〉 ↔ 〈𝑥, 0R〉 = 𝐴) | |
10 | 8, 9 | bitri 183 | . . . 4 ⊢ (∃𝑦 ∈ {0R}𝐴 = 〈𝑥, 𝑦〉 ↔ 〈𝑥, 0R〉 = 𝐴) |
11 | 10 | rexbii 2386 | . . 3 ⊢ (∃𝑥 ∈ R ∃𝑦 ∈ {0R}𝐴 = 〈𝑥, 𝑦〉 ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) |
12 | 3, 11 | bitri 183 | . 2 ⊢ (𝐴 ∈ (R × {0R}) ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) |
13 | 2, 12 | bitri 183 | 1 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1290 ∈ wcel 1439 ∃wrex 2361 {csn 3450 〈cop 3453 × cxp 4450 Rcnr 6917 0Rc0r 6918 ℝcr 7410 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-iinf 4416 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-tr 3943 df-eprel 4125 df-id 4129 df-po 4132 df-iso 4133 df-iord 4202 df-on 4204 df-suc 4207 df-iom 4419 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-rn 4463 df-res 4464 df-ima 4465 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-1st 5925 df-2nd 5926 df-recs 6084 df-irdg 6149 df-1o 6195 df-oadd 6199 df-omul 6200 df-er 6306 df-ec 6308 df-qs 6312 df-ni 6924 df-pli 6925 df-mi 6926 df-lti 6927 df-plpq 6964 df-mpq 6965 df-enq 6967 df-nqqs 6968 df-plqqs 6969 df-mqqs 6970 df-1nqqs 6971 df-rq 6972 df-ltnqqs 6973 df-inp 7086 df-i1p 7087 df-enr 7333 df-nr 7334 df-0r 7338 df-r 7421 |
This theorem is referenced by: elrealeu 7428 axaddrcl 7463 axmulrcl 7465 axprecex 7476 axpre-ltirr 7478 axpre-ltwlin 7479 axpre-lttrn 7480 axpre-apti 7481 axpre-ltadd 7482 axpre-mulgt0 7483 axpre-mulext 7484 axarch 7487 axcaucvglemres 7495 |
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