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Mirrors > Home > ILE Home > Th. List > prloc | Unicode version |
Description: A Dedekind cut is located. (Contributed by Jim Kingdon, 23-Oct-2019.) |
Ref | Expression |
---|---|
prloc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elinp 6664 | . . . 4 | |
2 | simpr3 946 | . . . 4 | |
3 | 1, 2 | sylbi 119 | . . 3 |
4 | 3 | adantr 270 | . 2 |
5 | simpr 108 | . 2 | |
6 | ltrelnq 6555 | . . . . . . 7 | |
7 | 6 | brel 4410 | . . . . . 6 |
8 | 7 | simpld 110 | . . . . 5 |
9 | 8 | adantl 271 | . . . 4 |
10 | simpr 108 | . . . . . . 7 | |
11 | 10 | breq1d 3795 | . . . . . 6 |
12 | 10 | eleq1d 2147 | . . . . . . 7 |
13 | 12 | orbi1d 737 | . . . . . 6 |
14 | 11, 13 | imbi12d 232 | . . . . 5 |
15 | 14 | ralbidv 2368 | . . . 4 |
16 | 9, 15 | rspcdv 2704 | . . 3 |
17 | 7 | simprd 112 | . . . . 5 |
18 | 17 | adantl 271 | . . . 4 |
19 | simpr 108 | . . . . . 6 | |
20 | 19 | breq2d 3797 | . . . . 5 |
21 | 19 | eleq1d 2147 | . . . . . 6 |
22 | 21 | orbi2d 736 | . . . . 5 |
23 | 20, 22 | imbi12d 232 | . . . 4 |
24 | 18, 23 | rspcdv 2704 | . . 3 |
25 | 16, 24 | syld 44 | . 2 |
26 | 4, 5, 25 | mp2d 46 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wo 661 w3a 919 wceq 1284 wcel 1433 wral 2348 wrex 2349 wss 2973 cop 3401 class class class wbr 3785 cnq 6470 cltq 6475 cnp 6481 |
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-pow 3948 ax-pr 3964 ax-un 4188 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 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-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-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-id 4048 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-qs 6135 df-ni 6494 df-nqqs 6538 df-ltnqqs 6543 df-inp 6656 |
This theorem is referenced by: prarloclem3step 6686 addnqprlemfl 6749 addnqprlemfu 6750 mullocprlem 6760 mulnqprlemfl 6765 mulnqprlemfu 6766 ltsopr 6786 ltexprlemloc 6797 addcanprleml 6804 addcanprlemu 6805 recexprlemloc 6821 cauappcvgprlemladdru 6846 cauappcvgprlemladdrl 6847 caucvgprlemladdrl 6868 |
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