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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 6572 | . . . 4 | |
2 | simpr3 912 | . . . 4 | |
3 | 1, 2 | sylbi 114 | . . 3 |
4 | 3 | adantr 261 | . 2 |
5 | simpr 103 | . 2 | |
6 | ltrelnq 6463 | . . . . . . 7 | |
7 | 6 | brel 4392 | . . . . . 6 |
8 | 7 | simpld 105 | . . . . 5 |
9 | 8 | adantl 262 | . . . 4 |
10 | simpr 103 | . . . . . . 7 | |
11 | 10 | breq1d 3774 | . . . . . 6 |
12 | 10 | eleq1d 2106 | . . . . . . 7 |
13 | 12 | orbi1d 705 | . . . . . 6 |
14 | 11, 13 | imbi12d 223 | . . . . 5 |
15 | 14 | ralbidv 2326 | . . . 4 |
16 | 9, 15 | rspcdv 2659 | . . 3 |
17 | 7 | simprd 107 | . . . . 5 |
18 | 17 | adantl 262 | . . . 4 |
19 | simpr 103 | . . . . . 6 | |
20 | 19 | breq2d 3776 | . . . . 5 |
21 | 19 | eleq1d 2106 | . . . . . 6 |
22 | 21 | orbi2d 704 | . . . . 5 |
23 | 20, 22 | imbi12d 223 | . . . 4 |
24 | 18, 23 | rspcdv 2659 | . . 3 |
25 | 16, 24 | syld 40 | . 2 |
26 | 4, 5, 25 | mp2d 41 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 wo 629 w3a 885 wceq 1243 wcel 1393 wral 2306 wrex 2307 wss 2917 cop 3378 class class class wbr 3764 cnq 6378 cltq 6383 cnp 6389 |
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-pow 3927 ax-pr 3944 ax-un 4170 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 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-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-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-id 4030 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-qs 6112 df-ni 6402 df-nqqs 6446 df-ltnqqs 6451 df-inp 6564 |
This theorem is referenced by: prarloclem3step 6594 addnqprlemfl 6657 addnqprlemfu 6658 mullocprlem 6668 mulnqprlemfl 6673 mulnqprlemfu 6674 ltsopr 6694 ltexprlemloc 6705 addcanprleml 6712 addcanprlemu 6713 recexprlemloc 6729 cauappcvgprlemladdru 6754 cauappcvgprlemladdrl 6755 caucvgprlemladdrl 6776 |
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