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Mirrors > Home > ILE Home > Th. List > nqprm | GIF version |
Description: A cut produced from a rational is inhabited. Lemma for nqprlu 7256. (Contributed by Jim Kingdon, 8-Dec-2019.) |
Ref | Expression |
---|---|
nqprm | ⊢ (𝐴 ∈ Q → (∃𝑞 ∈ Q 𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ∧ ∃𝑟 ∈ Q 𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nsmallnqq 7121 | . . 3 ⊢ (𝐴 ∈ Q → ∃𝑞 ∈ Q 𝑞 <Q 𝐴) | |
2 | vex 2644 | . . . . 5 ⊢ 𝑞 ∈ V | |
3 | breq1 3878 | . . . . 5 ⊢ (𝑥 = 𝑞 → (𝑥 <Q 𝐴 ↔ 𝑞 <Q 𝐴)) | |
4 | 2, 3 | elab 2782 | . . . 4 ⊢ (𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ↔ 𝑞 <Q 𝐴) |
5 | 4 | rexbii 2401 | . . 3 ⊢ (∃𝑞 ∈ Q 𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ↔ ∃𝑞 ∈ Q 𝑞 <Q 𝐴) |
6 | 1, 5 | sylibr 133 | . 2 ⊢ (𝐴 ∈ Q → ∃𝑞 ∈ Q 𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴}) |
7 | archnqq 7126 | . . . . 5 ⊢ (𝐴 ∈ Q → ∃𝑛 ∈ N 𝐴 <Q [〈𝑛, 1o〉] ~Q ) | |
8 | df-rex 2381 | . . . . 5 ⊢ (∃𝑛 ∈ N 𝐴 <Q [〈𝑛, 1o〉] ~Q ↔ ∃𝑛(𝑛 ∈ N ∧ 𝐴 <Q [〈𝑛, 1o〉] ~Q )) | |
9 | 7, 8 | sylib 121 | . . . 4 ⊢ (𝐴 ∈ Q → ∃𝑛(𝑛 ∈ N ∧ 𝐴 <Q [〈𝑛, 1o〉] ~Q )) |
10 | 1pi 7024 | . . . . . . . 8 ⊢ 1o ∈ N | |
11 | opelxpi 4509 | . . . . . . . . 9 ⊢ ((𝑛 ∈ N ∧ 1o ∈ N) → 〈𝑛, 1o〉 ∈ (N × N)) | |
12 | enqex 7069 | . . . . . . . . . 10 ⊢ ~Q ∈ V | |
13 | 12 | ecelqsi 6413 | . . . . . . . . 9 ⊢ (〈𝑛, 1o〉 ∈ (N × N) → [〈𝑛, 1o〉] ~Q ∈ ((N × N) / ~Q )) |
14 | 11, 13 | syl 14 | . . . . . . . 8 ⊢ ((𝑛 ∈ N ∧ 1o ∈ N) → [〈𝑛, 1o〉] ~Q ∈ ((N × N) / ~Q )) |
15 | 10, 14 | mpan2 419 | . . . . . . 7 ⊢ (𝑛 ∈ N → [〈𝑛, 1o〉] ~Q ∈ ((N × N) / ~Q )) |
16 | df-nqqs 7057 | . . . . . . 7 ⊢ Q = ((N × N) / ~Q ) | |
17 | 15, 16 | syl6eleqr 2193 | . . . . . 6 ⊢ (𝑛 ∈ N → [〈𝑛, 1o〉] ~Q ∈ Q) |
18 | breq2 3879 | . . . . . . 7 ⊢ (𝑟 = [〈𝑛, 1o〉] ~Q → (𝐴 <Q 𝑟 ↔ 𝐴 <Q [〈𝑛, 1o〉] ~Q )) | |
19 | 18 | rspcev 2744 | . . . . . 6 ⊢ (([〈𝑛, 1o〉] ~Q ∈ Q ∧ 𝐴 <Q [〈𝑛, 1o〉] ~Q ) → ∃𝑟 ∈ Q 𝐴 <Q 𝑟) |
20 | 17, 19 | sylan 279 | . . . . 5 ⊢ ((𝑛 ∈ N ∧ 𝐴 <Q [〈𝑛, 1o〉] ~Q ) → ∃𝑟 ∈ Q 𝐴 <Q 𝑟) |
21 | 20 | exlimiv 1545 | . . . 4 ⊢ (∃𝑛(𝑛 ∈ N ∧ 𝐴 <Q [〈𝑛, 1o〉] ~Q ) → ∃𝑟 ∈ Q 𝐴 <Q 𝑟) |
22 | 9, 21 | syl 14 | . . 3 ⊢ (𝐴 ∈ Q → ∃𝑟 ∈ Q 𝐴 <Q 𝑟) |
23 | vex 2644 | . . . . 5 ⊢ 𝑟 ∈ V | |
24 | breq2 3879 | . . . . 5 ⊢ (𝑥 = 𝑟 → (𝐴 <Q 𝑥 ↔ 𝐴 <Q 𝑟)) | |
25 | 23, 24 | elab 2782 | . . . 4 ⊢ (𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥} ↔ 𝐴 <Q 𝑟) |
26 | 25 | rexbii 2401 | . . 3 ⊢ (∃𝑟 ∈ Q 𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥} ↔ ∃𝑟 ∈ Q 𝐴 <Q 𝑟) |
27 | 22, 26 | sylibr 133 | . 2 ⊢ (𝐴 ∈ Q → ∃𝑟 ∈ Q 𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥}) |
28 | 6, 27 | jca 302 | 1 ⊢ (𝐴 ∈ Q → (∃𝑞 ∈ Q 𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ∧ ∃𝑟 ∈ Q 𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∃wex 1436 ∈ wcel 1448 {cab 2086 ∃wrex 2376 〈cop 3477 class class class wbr 3875 × cxp 4475 1oc1o 6236 [cec 6357 / cqs 6358 Ncnpi 6981 ~Q ceq 6988 Qcnq 6989 <Q cltq 6994 |
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 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-iinf 4440 |
This theorem depends on definitions: df-bi 116 df-dc 787 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-ral 2380 df-rex 2381 df-reu 2382 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-tr 3967 df-eprel 4149 df-id 4153 df-iord 4226 df-on 4228 df-suc 4231 df-iom 4443 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-recs 6132 df-irdg 6197 df-1o 6243 df-oadd 6247 df-omul 6248 df-er 6359 df-ec 6361 df-qs 6365 df-ni 7013 df-pli 7014 df-mi 7015 df-lti 7016 df-plpq 7053 df-mpq 7054 df-enq 7056 df-nqqs 7057 df-plqqs 7058 df-mqqs 7059 df-1nqqs 7060 df-rq 7061 df-ltnqqs 7062 |
This theorem is referenced by: nqprxx 7255 |
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