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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 7577. (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 7442 | . . 3 ⊢ (𝐴 ∈ Q → ∃𝑞 ∈ Q 𝑞 <Q 𝐴) | |
2 | vex 2755 | . . . . 5 ⊢ 𝑞 ∈ V | |
3 | breq1 4021 | . . . . 5 ⊢ (𝑥 = 𝑞 → (𝑥 <Q 𝐴 ↔ 𝑞 <Q 𝐴)) | |
4 | 2, 3 | elab 2896 | . . . 4 ⊢ (𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ↔ 𝑞 <Q 𝐴) |
5 | 4 | rexbii 2497 | . . 3 ⊢ (∃𝑞 ∈ Q 𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ↔ ∃𝑞 ∈ Q 𝑞 <Q 𝐴) |
6 | 1, 5 | sylibr 134 | . 2 ⊢ (𝐴 ∈ Q → ∃𝑞 ∈ Q 𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴}) |
7 | archnqq 7447 | . . . . 5 ⊢ (𝐴 ∈ Q → ∃𝑛 ∈ N 𝐴 <Q [〈𝑛, 1o〉] ~Q ) | |
8 | df-rex 2474 | . . . . 5 ⊢ (∃𝑛 ∈ N 𝐴 <Q [〈𝑛, 1o〉] ~Q ↔ ∃𝑛(𝑛 ∈ N ∧ 𝐴 <Q [〈𝑛, 1o〉] ~Q )) | |
9 | 7, 8 | sylib 122 | . . . 4 ⊢ (𝐴 ∈ Q → ∃𝑛(𝑛 ∈ N ∧ 𝐴 <Q [〈𝑛, 1o〉] ~Q )) |
10 | 1pi 7345 | . . . . . . . 8 ⊢ 1o ∈ N | |
11 | opelxpi 4676 | . . . . . . . . 9 ⊢ ((𝑛 ∈ N ∧ 1o ∈ N) → 〈𝑛, 1o〉 ∈ (N × N)) | |
12 | enqex 7390 | . . . . . . . . . 10 ⊢ ~Q ∈ V | |
13 | 12 | ecelqsi 6616 | . . . . . . . . 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 425 | . . . . . . 7 ⊢ (𝑛 ∈ N → [〈𝑛, 1o〉] ~Q ∈ ((N × N) / ~Q )) |
16 | df-nqqs 7378 | . . . . . . 7 ⊢ Q = ((N × N) / ~Q ) | |
17 | 15, 16 | eleqtrrdi 2283 | . . . . . 6 ⊢ (𝑛 ∈ N → [〈𝑛, 1o〉] ~Q ∈ Q) |
18 | breq2 4022 | . . . . . . 7 ⊢ (𝑟 = [〈𝑛, 1o〉] ~Q → (𝐴 <Q 𝑟 ↔ 𝐴 <Q [〈𝑛, 1o〉] ~Q )) | |
19 | 18 | rspcev 2856 | . . . . . 6 ⊢ (([〈𝑛, 1o〉] ~Q ∈ Q ∧ 𝐴 <Q [〈𝑛, 1o〉] ~Q ) → ∃𝑟 ∈ Q 𝐴 <Q 𝑟) |
20 | 17, 19 | sylan 283 | . . . . 5 ⊢ ((𝑛 ∈ N ∧ 𝐴 <Q [〈𝑛, 1o〉] ~Q ) → ∃𝑟 ∈ Q 𝐴 <Q 𝑟) |
21 | 20 | exlimiv 1609 | . . . 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 2755 | . . . . 5 ⊢ 𝑟 ∈ V | |
24 | breq2 4022 | . . . . 5 ⊢ (𝑥 = 𝑟 → (𝐴 <Q 𝑥 ↔ 𝐴 <Q 𝑟)) | |
25 | 23, 24 | elab 2896 | . . . 4 ⊢ (𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥} ↔ 𝐴 <Q 𝑟) |
26 | 25 | rexbii 2497 | . . 3 ⊢ (∃𝑟 ∈ Q 𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥} ↔ ∃𝑟 ∈ Q 𝐴 <Q 𝑟) |
27 | 22, 26 | sylibr 134 | . 2 ⊢ (𝐴 ∈ Q → ∃𝑟 ∈ Q 𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥}) |
28 | 6, 27 | jca 306 | 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 104 ∃wex 1503 ∈ wcel 2160 {cab 2175 ∃wrex 2469 〈cop 3610 class class class wbr 4018 × cxp 4642 1oc1o 6435 [cec 6558 / cqs 6559 Ncnpi 7302 ~Q ceq 7309 Qcnq 7310 <Q cltq 7315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-eprel 4307 df-id 4311 df-iord 4384 df-on 4386 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-ov 5900 df-oprab 5901 df-mpo 5902 df-1st 6166 df-2nd 6167 df-recs 6331 df-irdg 6396 df-1o 6442 df-oadd 6446 df-omul 6447 df-er 6560 df-ec 6562 df-qs 6566 df-ni 7334 df-pli 7335 df-mi 7336 df-lti 7337 df-plpq 7374 df-mpq 7375 df-enq 7377 df-nqqs 7378 df-plqqs 7379 df-mqqs 7380 df-1nqqs 7381 df-rq 7382 df-ltnqqs 7383 |
This theorem is referenced by: nqprxx 7576 |
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