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Mirrors > Home > ILE Home > Th. List > zlelttric | Unicode version |
Description: Trichotomy law. (Contributed by Jim Kingdon, 17-Apr-2020.) |
Ref | Expression |
---|---|
zlelttric |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zre 9061 | . . 3 | |
2 | zre 9061 | . . 3 | |
3 | 1, 2 | anim12i 336 | . 2 |
4 | ztri3or 9100 | . 2 | |
5 | ltle 7854 | . . . 4 | |
6 | orc 701 | . . . 4 | |
7 | 5, 6 | syl6 33 | . . 3 |
8 | eqle 7858 | . . . . . 6 | |
9 | 8 | ex 114 | . . . . 5 |
10 | 9 | adantr 274 | . . . 4 |
11 | 10, 6 | syl6 33 | . . 3 |
12 | olc 700 | . . . 4 | |
13 | 12 | a1i 9 | . . 3 |
14 | 7, 11, 13 | 3jaod 1282 | . 2 |
15 | 3, 4, 14 | sylc 62 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 697 w3o 961 wceq 1331 wcel 1480 class class class wbr 3929 cr 7622 clt 7803 cle 7804 cz 9057 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-addcom 7723 ax-addass 7725 ax-distr 7727 ax-i2m1 7728 ax-0lt1 7729 ax-0id 7731 ax-rnegex 7732 ax-cnre 7734 ax-pre-ltirr 7735 ax-pre-ltwlin 7736 ax-pre-lttrn 7737 ax-pre-ltadd 7739 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-sub 7938 df-neg 7939 df-inn 8724 df-n0 8981 df-z 9058 |
This theorem is referenced by: btwnapz 9184 eluzdc 9407 fzsplit2 9833 uzsplit 9875 fzospliti 9956 fzouzsplit 9959 faclbnd 10490 resqrexlemoverl 10796 fisumrev2 11218 dvdslelemd 11544 dvdsle 11545 sqrt2irrap 11861 uzdcinzz 13008 |
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