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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 6991 | . . 3 ⊢ ℝ = (R × {0R}) | |
2 | 1 | eleq2i 2145 | . 2 ⊢ (𝐴 ∈ ℝ ↔ 𝐴 ∈ (R × {0R})) |
3 | elxp2 4381 | . . 3 ⊢ (𝐴 ∈ (R × {0R}) ↔ ∃𝑥 ∈ R ∃𝑦 ∈ {0R}𝐴 = 〈𝑥, 𝑦〉) | |
4 | 0r 6927 | . . . . . . 7 ⊢ 0R ∈ R | |
5 | 4 | elexi 2611 | . . . . . 6 ⊢ 0R ∈ V |
6 | opeq2 3571 | . . . . . . 7 ⊢ (𝑦 = 0R → 〈𝑥, 𝑦〉 = 〈𝑥, 0R〉) | |
7 | 6 | eqeq2d 2092 | . . . . . 6 ⊢ (𝑦 = 0R → (𝐴 = 〈𝑥, 𝑦〉 ↔ 𝐴 = 〈𝑥, 0R〉)) |
8 | 5, 7 | rexsn 3437 | . . . . 5 ⊢ (∃𝑦 ∈ {0R}𝐴 = 〈𝑥, 𝑦〉 ↔ 𝐴 = 〈𝑥, 0R〉) |
9 | eqcom 2083 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 0R〉 ↔ 〈𝑥, 0R〉 = 𝐴) | |
10 | 8, 9 | bitri 182 | . . . 4 ⊢ (∃𝑦 ∈ {0R}𝐴 = 〈𝑥, 𝑦〉 ↔ 〈𝑥, 0R〉 = 𝐴) |
11 | 10 | rexbii 2373 | . . 3 ⊢ (∃𝑥 ∈ R ∃𝑦 ∈ {0R}𝐴 = 〈𝑥, 𝑦〉 ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) |
12 | 3, 11 | bitri 182 | . 2 ⊢ (𝐴 ∈ (R × {0R}) ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) |
13 | 2, 12 | bitri 182 | 1 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 = wceq 1284 ∈ wcel 1433 ∃wrex 2349 {csn 3398 〈cop 3401 × cxp 4361 Rcnr 6487 0Rc0r 6488 ℝcr 6980 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-pli 6495 df-mi 6496 df-lti 6497 df-plpq 6534 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-plqqs 6539 df-mqqs 6540 df-1nqqs 6541 df-rq 6542 df-ltnqqs 6543 df-inp 6656 df-i1p 6657 df-enr 6903 df-nr 6904 df-0r 6908 df-r 6991 |
This theorem is referenced by: elrealeu 6998 axaddrcl 7033 axmulrcl 7035 axprecex 7046 axpre-ltirr 7048 axpre-ltwlin 7049 axpre-lttrn 7050 axpre-apti 7051 axpre-ltadd 7052 axpre-mulgt0 7053 axpre-mulext 7054 axarch 7057 axcaucvglemres 7065 |
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