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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 7479. (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 7344 | . . 3 ⊢ (𝐴 ∈ Q → ∃𝑞 ∈ Q 𝑞 <Q 𝐴) | |
2 | vex 2724 | . . . . 5 ⊢ 𝑞 ∈ V | |
3 | breq1 3979 | . . . . 5 ⊢ (𝑥 = 𝑞 → (𝑥 <Q 𝐴 ↔ 𝑞 <Q 𝐴)) | |
4 | 2, 3 | elab 2865 | . . . 4 ⊢ (𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ↔ 𝑞 <Q 𝐴) |
5 | 4 | rexbii 2471 | . . 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 7349 | . . . . 5 ⊢ (𝐴 ∈ Q → ∃𝑛 ∈ N 𝐴 <Q [〈𝑛, 1o〉] ~Q ) | |
8 | df-rex 2448 | . . . . 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 7247 | . . . . . . . 8 ⊢ 1o ∈ N | |
11 | opelxpi 4630 | . . . . . . . . 9 ⊢ ((𝑛 ∈ N ∧ 1o ∈ N) → 〈𝑛, 1o〉 ∈ (N × N)) | |
12 | enqex 7292 | . . . . . . . . . 10 ⊢ ~Q ∈ V | |
13 | 12 | ecelqsi 6546 | . . . . . . . . 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 422 | . . . . . . 7 ⊢ (𝑛 ∈ N → [〈𝑛, 1o〉] ~Q ∈ ((N × N) / ~Q )) |
16 | df-nqqs 7280 | . . . . . . 7 ⊢ Q = ((N × N) / ~Q ) | |
17 | 15, 16 | eleqtrrdi 2258 | . . . . . 6 ⊢ (𝑛 ∈ N → [〈𝑛, 1o〉] ~Q ∈ Q) |
18 | breq2 3980 | . . . . . . 7 ⊢ (𝑟 = [〈𝑛, 1o〉] ~Q → (𝐴 <Q 𝑟 ↔ 𝐴 <Q [〈𝑛, 1o〉] ~Q )) | |
19 | 18 | rspcev 2825 | . . . . . 6 ⊢ (([〈𝑛, 1o〉] ~Q ∈ Q ∧ 𝐴 <Q [〈𝑛, 1o〉] ~Q ) → ∃𝑟 ∈ Q 𝐴 <Q 𝑟) |
20 | 17, 19 | sylan 281 | . . . . 5 ⊢ ((𝑛 ∈ N ∧ 𝐴 <Q [〈𝑛, 1o〉] ~Q ) → ∃𝑟 ∈ Q 𝐴 <Q 𝑟) |
21 | 20 | exlimiv 1585 | . . . 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 2724 | . . . . 5 ⊢ 𝑟 ∈ V | |
24 | breq2 3980 | . . . . 5 ⊢ (𝑥 = 𝑟 → (𝐴 <Q 𝑥 ↔ 𝐴 <Q 𝑟)) | |
25 | 23, 24 | elab 2865 | . . . 4 ⊢ (𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥} ↔ 𝐴 <Q 𝑟) |
26 | 25 | rexbii 2471 | . . 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 304 | 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 1479 ∈ wcel 2135 {cab 2150 ∃wrex 2443 〈cop 3573 class class class wbr 3976 × cxp 4596 1oc1o 6368 [cec 6490 / cqs 6491 Ncnpi 7204 ~Q ceq 7211 Qcnq 7212 <Q cltq 7217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-eprel 4261 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-1o 6375 df-oadd 6379 df-omul 6380 df-er 6492 df-ec 6494 df-qs 6498 df-ni 7236 df-pli 7237 df-mi 7238 df-lti 7239 df-plpq 7276 df-mpq 7277 df-enq 7279 df-nqqs 7280 df-plqqs 7281 df-mqqs 7282 df-1nqqs 7283 df-rq 7284 df-ltnqqs 7285 |
This theorem is referenced by: nqprxx 7478 |
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