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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 7443 | . . . . 5 | |
2 | elinp 7436 | . . . . . . . 8 | |
3 | simpr1l 1049 | . . . . . . . 8 | |
4 | 2, 3 | sylbi 120 | . . . . . . 7 |
5 | eleq1 2233 | . . . . . . . . 9 | |
6 | breq1 3992 | . . . . . . . . . . 11 | |
7 | 6 | anbi1d 462 | . . . . . . . . . 10 |
8 | 7 | rexbidv 2471 | . . . . . . . . 9 |
9 | 5, 8 | bibi12d 234 | . . . . . . . 8 |
10 | 9 | rspcv 2830 | . . . . . . 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 2454 | . . . 4 | |
16 | 14, 15 | sylib 121 | . . 3 |
17 | ltrelnq 7327 | . . . . . . . . 9 | |
18 | 17 | brel 4663 | . . . . . . . 8 |
19 | 18 | simprd 113 | . . . . . . 7 |
20 | 19 | pm4.71ri 390 | . . . . . 6 |
21 | 20 | anbi1i 455 | . . . . 5 |
22 | ancom 264 | . . . . 5 | |
23 | anass 399 | . . . . 5 | |
24 | 21, 22, 23 | 3bitr3i 209 | . . . 4 |
25 | 24 | exbii 1598 | . . 3 |
26 | 16, 25 | sylibr 133 | . 2 |
27 | df-rex 2454 | . 2 | |
28 | 26, 27 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 703 w3a 973 wceq 1348 wex 1485 wcel 2141 wral 2448 wrex 2449 wss 3121 cop 3586 class class class wbr 3989 cnq 7242 cltq 7247 cnp 7253 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-qs 6519 df-ni 7266 df-nqqs 7310 df-ltnqqs 7315 df-inp 7428 |
This theorem is referenced by: prnmaddl 7452 genprndl 7483 nqprl 7513 1idprl 7552 ltsopr 7558 ltexprlemm 7562 ltexprlemopl 7563 recexprlemloc 7593 recexprlem1ssl 7595 aptiprleml 7601 caucvgprprlemopl 7659 suplocexprlemrl 7679 |
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