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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 6737. (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 6602 | . . 3 ⊢ (𝐴 ∈ Q → ∃𝑞 ∈ Q 𝑞 <Q 𝐴) | |
2 | vex 2604 | . . . . 5 ⊢ 𝑞 ∈ V | |
3 | breq1 3788 | . . . . 5 ⊢ (𝑥 = 𝑞 → (𝑥 <Q 𝐴 ↔ 𝑞 <Q 𝐴)) | |
4 | 2, 3 | elab 2738 | . . . 4 ⊢ (𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ↔ 𝑞 <Q 𝐴) |
5 | 4 | rexbii 2373 | . . 3 ⊢ (∃𝑞 ∈ Q 𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ↔ ∃𝑞 ∈ Q 𝑞 <Q 𝐴) |
6 | 1, 5 | sylibr 132 | . 2 ⊢ (𝐴 ∈ Q → ∃𝑞 ∈ Q 𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴}) |
7 | archnqq 6607 | . . . . 5 ⊢ (𝐴 ∈ Q → ∃𝑛 ∈ N 𝐴 <Q [〈𝑛, 1𝑜〉] ~Q ) | |
8 | df-rex 2354 | . . . . 5 ⊢ (∃𝑛 ∈ N 𝐴 <Q [〈𝑛, 1𝑜〉] ~Q ↔ ∃𝑛(𝑛 ∈ N ∧ 𝐴 <Q [〈𝑛, 1𝑜〉] ~Q )) | |
9 | 7, 8 | sylib 120 | . . . 4 ⊢ (𝐴 ∈ Q → ∃𝑛(𝑛 ∈ N ∧ 𝐴 <Q [〈𝑛, 1𝑜〉] ~Q )) |
10 | 1pi 6505 | . . . . . . . 8 ⊢ 1𝑜 ∈ N | |
11 | opelxpi 4394 | . . . . . . . . 9 ⊢ ((𝑛 ∈ N ∧ 1𝑜 ∈ N) → 〈𝑛, 1𝑜〉 ∈ (N × N)) | |
12 | enqex 6550 | . . . . . . . . . 10 ⊢ ~Q ∈ V | |
13 | 12 | ecelqsi 6183 | . . . . . . . . 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 415 | . . . . . . 7 ⊢ (𝑛 ∈ N → [〈𝑛, 1𝑜〉] ~Q ∈ ((N × N) / ~Q )) |
16 | df-nqqs 6538 | . . . . . . 7 ⊢ Q = ((N × N) / ~Q ) | |
17 | 15, 16 | syl6eleqr 2172 | . . . . . 6 ⊢ (𝑛 ∈ N → [〈𝑛, 1𝑜〉] ~Q ∈ Q) |
18 | breq2 3789 | . . . . . . 7 ⊢ (𝑟 = [〈𝑛, 1𝑜〉] ~Q → (𝐴 <Q 𝑟 ↔ 𝐴 <Q [〈𝑛, 1𝑜〉] ~Q )) | |
19 | 18 | rspcev 2701 | . . . . . 6 ⊢ (([〈𝑛, 1𝑜〉] ~Q ∈ Q ∧ 𝐴 <Q [〈𝑛, 1𝑜〉] ~Q ) → ∃𝑟 ∈ Q 𝐴 <Q 𝑟) |
20 | 17, 19 | sylan 277 | . . . . 5 ⊢ ((𝑛 ∈ N ∧ 𝐴 <Q [〈𝑛, 1𝑜〉] ~Q ) → ∃𝑟 ∈ Q 𝐴 <Q 𝑟) |
21 | 20 | exlimiv 1529 | . . . 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 2604 | . . . . 5 ⊢ 𝑟 ∈ V | |
24 | breq2 3789 | . . . . 5 ⊢ (𝑥 = 𝑟 → (𝐴 <Q 𝑥 ↔ 𝐴 <Q 𝑟)) | |
25 | 23, 24 | elab 2738 | . . . 4 ⊢ (𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥} ↔ 𝐴 <Q 𝑟) |
26 | 25 | rexbii 2373 | . . 3 ⊢ (∃𝑟 ∈ Q 𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥} ↔ ∃𝑟 ∈ Q 𝐴 <Q 𝑟) |
27 | 22, 26 | sylibr 132 | . 2 ⊢ (𝐴 ∈ Q → ∃𝑟 ∈ Q 𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥}) |
28 | 6, 27 | jca 300 | 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 102 ∃wex 1421 ∈ wcel 1433 {cab 2067 ∃wrex 2349 〈cop 3401 class class class wbr 3785 × cxp 4361 1𝑜c1o 6017 [cec 6127 / cqs 6128 Ncnpi 6462 ~Q ceq 6469 Qcnq 6470 <Q cltq 6475 |
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-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 |
This theorem is referenced by: nqprxx 6736 |
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