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Mirrors > Home > ILE Home > Th. List > nqprloc | Unicode version |
Description: A cut produced from a rational is located. Lemma for nqprlu 6645. (Contributed by Jim Kingdon, 8-Dec-2019.) |
Ref | Expression |
---|---|
nqprloc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nqtri3or 6494 | . . . . . . 7 | |
2 | 1 | ancoms 255 | . . . . . 6 |
3 | 2 | ad2antrr 457 | . . . . 5 |
4 | vex 2560 | . . . . . . . . . 10 | |
5 | breq1 3767 | . . . . . . . . . 10 | |
6 | 4, 5 | elab 2687 | . . . . . . . . 9 |
7 | 6 | biimpri 124 | . . . . . . . 8 |
8 | 7 | orcd 652 | . . . . . . 7 |
9 | 8 | a1i 9 | . . . . . 6 |
10 | simpr 103 | . . . . . . . 8 | |
11 | breq1 3767 | . . . . . . . 8 | |
12 | 10, 11 | syl5ibcom 144 | . . . . . . 7 |
13 | vex 2560 | . . . . . . . . 9 | |
14 | breq2 3768 | . . . . . . . . 9 | |
15 | 13, 14 | elab 2687 | . . . . . . . 8 |
16 | olc 632 | . . . . . . . 8 | |
17 | 15, 16 | sylbir 125 | . . . . . . 7 |
18 | 12, 17 | syl6 29 | . . . . . 6 |
19 | ltsonq 6496 | . . . . . . . . . 10 | |
20 | ltrelnq 6463 | . . . . . . . . . 10 | |
21 | 19, 20 | sotri 4720 | . . . . . . . . 9 |
22 | 21, 17 | syl 14 | . . . . . . . 8 |
23 | 22 | expcom 109 | . . . . . . 7 |
24 | 23 | adantl 262 | . . . . . 6 |
25 | 9, 18, 24 | 3jaod 1199 | . . . . 5 |
26 | 3, 25 | mpd 13 | . . . 4 |
27 | 26 | ex 108 | . . 3 |
28 | 27 | ralrimiva 2392 | . 2 |
29 | 28 | ralrimiva 2392 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wo 629 w3o 884 wceq 1243 wcel 1393 cab 2026 wral 2306 class class class wbr 3764 cnq 6378 cltq 6383 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-mi 6404 df-lti 6405 df-enq 6445 df-nqqs 6446 df-ltnqqs 6451 |
This theorem is referenced by: nqprxx 6644 |
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