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Mirrors > Home > ILE Home > Th. List > zltnle | Unicode version |
Description: 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 14-Mar-2020.) |
Ref | Expression |
---|---|
zltnle |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zre 9209 | . . . . 5 | |
2 | zre 9209 | . . . . 5 | |
3 | lenlt 7988 | . . . . 5 | |
4 | 1, 2, 3 | syl2anr 288 | . . . 4 |
5 | 4 | biimpd 143 | . . 3 |
6 | 5 | con2d 619 | . 2 |
7 | ztri3or 9248 | . . 3 | |
8 | ax-1 6 | . . . . 5 | |
9 | 8 | a1i 9 | . . . 4 |
10 | eqcom 2172 | . . . . . . . . 9 | |
11 | eqle 8004 | . . . . . . . . 9 | |
12 | 10, 11 | sylan2b 285 | . . . . . . . 8 |
13 | 12 | ex 114 | . . . . . . 7 |
14 | 13 | adantl 275 | . . . . . 6 |
15 | 1, 14 | sylan2 284 | . . . . 5 |
16 | pm2.24 616 | . . . . 5 | |
17 | 15, 16 | syl6 33 | . . . 4 |
18 | ltle 8000 | . . . . . 6 | |
19 | 1, 2, 18 | syl2anr 288 | . . . . 5 |
20 | 19, 16 | syl6 33 | . . . 4 |
21 | 9, 17, 20 | 3jaod 1299 | . . 3 |
22 | 7, 21 | mpd 13 | . 2 |
23 | 6, 22 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3o 972 wceq 1348 wcel 2141 class class class wbr 3987 cr 7766 clt 7947 cle 7948 cz 9205 |
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-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-addcom 7867 ax-addass 7869 ax-distr 7871 ax-i2m1 7872 ax-0lt1 7873 ax-0id 7875 ax-rnegex 7876 ax-cnre 7878 ax-pre-ltirr 7879 ax-pre-ltwlin 7880 ax-pre-lttrn 7881 ax-pre-ltadd 7883 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 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-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-sub 8085 df-neg 8086 df-inn 8872 df-n0 9129 df-z 9206 |
This theorem is referenced by: znnnlt1 9253 nn0n0n1ge2b 9284 eluzdc 9562 fzdcel 9989 fzn 9991 fzpreddisj 10020 fzp1disj 10029 fzneuz 10050 fznuz 10051 uznfz 10052 fzp1nel 10053 difelfznle 10084 fzodisj 10127 exfzdc 10189 modfzo0difsn 10344 fzfig 10379 iseqf1olemqk 10443 exp3val 10471 facdiv 10665 bcval5 10690 zfz1isolemiso 10767 2zsupmax 11182 2zinfmin 11199 summodclem3 11336 fprodntrivap 11540 alzdvds 11807 fzm1ndvds 11809 fzo0dvdseq 11810 n2dvds1 11864 dvdsbnd 11904 algcvgblem 11996 prmndvdsfaclt 12103 odzdvds 12192 pcprendvds 12237 pcdvdsb 12266 pc2dvds 12276 pcmpt 12288 pockthg 12302 prmunb 12307 1arith 12312 lgsdilem2 13696 uzdcinzz 13798 |
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