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Mirrors > Home > ILE Home > Th. List > prnmaxl | Unicode version |
Description: A lower cut has no largest member. (Contributed by Jim Kingdon, 29-Sep-2019.) |
Ref | Expression |
---|---|
prnmaxl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elprnql 6579 | . . . . 5 | |
2 | elinp 6572 | . . . . . . . 8 | |
3 | simpr1l 961 | . . . . . . . 8 | |
4 | 2, 3 | sylbi 114 | . . . . . . 7 |
5 | eleq1 2100 | . . . . . . . . 9 | |
6 | breq1 3767 | . . . . . . . . . . 11 | |
7 | 6 | anbi1d 438 | . . . . . . . . . 10 |
8 | 7 | rexbidv 2327 | . . . . . . . . 9 |
9 | 5, 8 | bibi12d 224 | . . . . . . . 8 |
10 | 9 | rspcv 2652 | . . . . . . 7 |
11 | bi1 111 | . . . . . . 7 | |
12 | 4, 10, 11 | syl56 30 | . . . . . 6 |
13 | 12 | impd 242 | . . . . 5 |
14 | 1, 13 | mpcom 32 | . . . 4 |
15 | df-rex 2312 | . . . 4 | |
16 | 14, 15 | sylib 127 | . . 3 |
17 | ltrelnq 6463 | . . . . . . . . 9 | |
18 | 17 | brel 4392 | . . . . . . . 8 |
19 | 18 | simprd 107 | . . . . . . 7 |
20 | 19 | pm4.71ri 372 | . . . . . 6 |
21 | 20 | anbi1i 431 | . . . . 5 |
22 | ancom 253 | . . . . 5 | |
23 | anass 381 | . . . . 5 | |
24 | 21, 22, 23 | 3bitr3i 199 | . . . 4 |
25 | 24 | exbii 1496 | . . 3 |
26 | 16, 25 | sylibr 137 | . 2 |
27 | df-rex 2312 | . 2 | |
28 | 26, 27 | sylibr 137 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 wo 629 w3a 885 wceq 1243 wex 1381 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: prnmaddl 6588 genprndl 6619 nqprl 6649 1idprl 6688 ltsopr 6694 ltexprlemm 6698 ltexprlemopl 6699 recexprlemloc 6729 recexprlem1ssl 6731 aptiprleml 6737 caucvgprprlemopl 6795 |
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