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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 7413 | . . . . 5 | |
2 | elinp 7406 | . . . . . . . 8 | |
3 | simpr1l 1043 | . . . . . . . 8 | |
4 | 2, 3 | sylbi 120 | . . . . . . 7 |
5 | eleq1 2227 | . . . . . . . . 9 | |
6 | breq1 3979 | . . . . . . . . . . 11 | |
7 | 6 | anbi1d 461 | . . . . . . . . . 10 |
8 | 7 | rexbidv 2465 | . . . . . . . . 9 |
9 | 5, 8 | bibi12d 234 | . . . . . . . 8 |
10 | 9 | rspcv 2821 | . . . . . . 7 |
11 | biimp 117 | . . . . . . 7 | |
12 | 4, 10, 11 | syl56 34 | . . . . . 6 |
13 | 12 | impd 252 | . . . . 5 |
14 | 1, 13 | mpcom 36 | . . . 4 |
15 | df-rex 2448 | . . . 4 | |
16 | 14, 15 | sylib 121 | . . 3 |
17 | ltrelnq 7297 | . . . . . . . . 9 | |
18 | 17 | brel 4650 | . . . . . . . 8 |
19 | 18 | simprd 113 | . . . . . . 7 |
20 | 19 | pm4.71ri 390 | . . . . . 6 |
21 | 20 | anbi1i 454 | . . . . 5 |
22 | ancom 264 | . . . . 5 | |
23 | anass 399 | . . . . 5 | |
24 | 21, 22, 23 | 3bitr3i 209 | . . . 4 |
25 | 24 | exbii 1592 | . . 3 |
26 | 16, 25 | sylibr 133 | . 2 |
27 | df-rex 2448 | . 2 | |
28 | 26, 27 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 w3a 967 wceq 1342 wex 1479 wcel 2135 wral 2442 wrex 2443 wss 3111 cop 3573 class class class wbr 3976 cnq 7212 cltq 7217 cnp 7223 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-qs 6498 df-ni 7236 df-nqqs 7280 df-ltnqqs 7285 df-inp 7398 |
This theorem is referenced by: prnmaddl 7422 genprndl 7453 nqprl 7483 1idprl 7522 ltsopr 7528 ltexprlemm 7532 ltexprlemopl 7533 recexprlemloc 7563 recexprlem1ssl 7565 aptiprleml 7571 caucvgprprlemopl 7629 suplocexprlemrl 7649 |
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