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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 1151 | . . 3 | |
2 | 1 | adantl 275 | . 2 |
3 | 3mix3 1153 | . . . 4 | |
4 | 3 | adantl 275 | . . 3 |
5 | simpr 109 | . . . . . . 7 | |
6 | simpl2 986 | . . . . . . . . 9 | |
7 | 6 | zred 9292 | . . . . . . . 8 |
8 | simpl1 985 | . . . . . . . . 9 | |
9 | 8 | zred 9292 | . . . . . . . 8 |
10 | 7, 9 | lenltd 7998 | . . . . . . 7 |
11 | 5, 10 | mpbird 166 | . . . . . 6 |
12 | 11 | adantr 274 | . . . . 5 |
13 | simpr 109 | . . . . . 6 | |
14 | 9 | adantr 274 | . . . . . . 7 |
15 | simpll3 1023 | . . . . . . . 8 | |
16 | 15 | zred 9292 | . . . . . . 7 |
17 | 14, 16 | lenltd 7998 | . . . . . 6 |
18 | 13, 17 | mpbird 166 | . . . . 5 |
19 | elfz 9925 | . . . . . . 7 | |
20 | 19 | adantr 274 | . . . . . 6 |
21 | 20 | adantr 274 | . . . . 5 |
22 | 12, 18, 21 | mpbir2and 929 | . . . 4 |
23 | 22 | 3mix2d 1158 | . . 3 |
24 | zdclt 9247 | . . . . . . 7 DECID | |
25 | 24 | ancoms 266 | . . . . . 6 DECID |
26 | 25 | 3adant2 1001 | . . . . 5 DECID |
27 | 26 | adantr 274 | . . . 4 DECID |
28 | df-dc 821 | . . . 4 DECID | |
29 | 27, 28 | sylib 121 | . . 3 |
30 | 4, 23, 29 | mpjaodan 788 | . 2 |
31 | zdclt 9247 | . . . 4 DECID | |
32 | 31 | 3adant3 1002 | . . 3 DECID |
33 | df-dc 821 | . . 3 DECID | |
34 | 32, 33 | sylib 121 | . 2 |
35 | 2, 30, 34 | mpjaodan 788 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 DECID wdc 820 w3o 962 w3a 963 wcel 2128 class class class wbr 3967 (class class class)co 5827 cr 7734 clt 7915 cle 7916 cz 9173 cfz 9919 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4085 ax-pow 4138 ax-pr 4172 ax-un 4396 ax-setind 4499 ax-cnex 7826 ax-resscn 7827 ax-1cn 7828 ax-1re 7829 ax-icn 7830 ax-addcl 7831 ax-addrcl 7832 ax-mulcl 7833 ax-addcom 7835 ax-addass 7837 ax-distr 7839 ax-i2m1 7840 ax-0lt1 7841 ax-0id 7843 ax-rnegex 7844 ax-cnre 7846 ax-pre-ltirr 7847 ax-pre-ltwlin 7848 ax-pre-lttrn 7849 ax-pre-ltadd 7851 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-br 3968 df-opab 4029 df-id 4256 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-iota 5138 df-fun 5175 df-fv 5181 df-riota 5783 df-ov 5830 df-oprab 5831 df-mpo 5832 df-pnf 7917 df-mnf 7918 df-xr 7919 df-ltxr 7920 df-le 7921 df-sub 8053 df-neg 8054 df-inn 8840 df-n0 9097 df-z 9174 df-fz 9920 |
This theorem is referenced by: fzdcel 9949 hashfiv01gt1 10668 |
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