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Mirrors > Home > ILE Home > Th. List > fztri3or | Unicode version |
Description: Trichotomy in terms of a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.) |
Ref | Expression |
---|---|
fztri3or |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3mix1 1150 | . . 3 | |
2 | 1 | adantl 275 | . 2 |
3 | 3mix3 1152 | . . . 4 | |
4 | 3 | adantl 275 | . . 3 |
5 | simpr 109 | . . . . . . 7 | |
6 | simpl2 985 | . . . . . . . . 9 | |
7 | 6 | zred 9180 | . . . . . . . 8 |
8 | simpl1 984 | . . . . . . . . 9 | |
9 | 8 | zred 9180 | . . . . . . . 8 |
10 | 7, 9 | lenltd 7887 | . . . . . . 7 |
11 | 5, 10 | mpbird 166 | . . . . . 6 |
12 | 11 | adantr 274 | . . . . 5 |
13 | simpr 109 | . . . . . 6 | |
14 | 9 | adantr 274 | . . . . . . 7 |
15 | simpll3 1022 | . . . . . . . 8 | |
16 | 15 | zred 9180 | . . . . . . 7 |
17 | 14, 16 | lenltd 7887 | . . . . . 6 |
18 | 13, 17 | mpbird 166 | . . . . 5 |
19 | elfz 9803 | . . . . . . 7 | |
20 | 19 | adantr 274 | . . . . . 6 |
21 | 20 | adantr 274 | . . . . 5 |
22 | 12, 18, 21 | mpbir2and 928 | . . . 4 |
23 | 22 | 3mix2d 1157 | . . 3 |
24 | zdclt 9135 | . . . . . . 7 DECID | |
25 | 24 | ancoms 266 | . . . . . 6 DECID |
26 | 25 | 3adant2 1000 | . . . . 5 DECID |
27 | 26 | adantr 274 | . . . 4 DECID |
28 | df-dc 820 | . . . 4 DECID | |
29 | 27, 28 | sylib 121 | . . 3 |
30 | 4, 23, 29 | mpjaodan 787 | . 2 |
31 | zdclt 9135 | . . . 4 DECID | |
32 | 31 | 3adant3 1001 | . . 3 DECID |
33 | df-dc 820 | . . 3 DECID | |
34 | 32, 33 | sylib 121 | . 2 |
35 | 2, 30, 34 | mpjaodan 787 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 DECID wdc 819 w3o 961 w3a 962 wcel 1480 class class class wbr 3929 (class class class)co 5774 cr 7626 clt 7807 cle 7808 cz 9061 cfz 9797 |
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 7718 ax-resscn 7719 ax-1cn 7720 ax-1re 7721 ax-icn 7722 ax-addcl 7723 ax-addrcl 7724 ax-mulcl 7725 ax-addcom 7727 ax-addass 7729 ax-distr 7731 ax-i2m1 7732 ax-0lt1 7733 ax-0id 7735 ax-rnegex 7736 ax-cnre 7738 ax-pre-ltirr 7739 ax-pre-ltwlin 7740 ax-pre-lttrn 7741 ax-pre-ltadd 7743 |
This theorem depends on definitions: df-bi 116 df-dc 820 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 7809 df-mnf 7810 df-xr 7811 df-ltxr 7812 df-le 7813 df-sub 7942 df-neg 7943 df-inn 8728 df-n0 8985 df-z 9062 df-fz 9798 |
This theorem is referenced by: fzdcel 9827 hashfiv01gt1 10535 |
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