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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zre 9286 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | zre 9286 |
. . . . 5
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3 | lenlt 8062 |
. . . . 5
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4 | 1, 2, 3 | syl2anr 290 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | 4 | biimpd 144 |
. . 3
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6 | 5 | con2d 625 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7 | ztri3or 9325 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | ax-1 6 |
. . . . 5
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9 | 8 | a1i 9 |
. . . 4
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10 | eqcom 2191 |
. . . . . . . . 9
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11 | eqle 8078 |
. . . . . . . . 9
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12 | 10, 11 | sylan2b 287 |
. . . . . . . 8
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13 | 12 | ex 115 |
. . . . . . 7
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14 | 13 | adantl 277 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 1, 14 | sylan2 286 |
. . . . 5
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16 | pm2.24 622 |
. . . . 5
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17 | 15, 16 | syl6 33 |
. . . 4
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18 | ltle 8074 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | 1, 2, 18 | syl2anr 290 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | 19, 16 | syl6 33 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | 9, 17, 20 | 3jaod 1315 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 7, 21 | mpd 13 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 6, 22 | impbid 129 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7931 ax-resscn 7932 ax-1cn 7933 ax-1re 7934 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-addcom 7940 ax-addass 7942 ax-distr 7944 ax-i2m1 7945 ax-0lt1 7946 ax-0id 7948 ax-rnegex 7949 ax-cnre 7951 ax-pre-ltirr 7952 ax-pre-ltwlin 7953 ax-pre-lttrn 7954 ax-pre-ltadd 7956 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5851 df-ov 5898 df-oprab 5899 df-mpo 5900 df-pnf 8023 df-mnf 8024 df-xr 8025 df-ltxr 8026 df-le 8027 df-sub 8159 df-neg 8160 df-inn 8949 df-n0 9206 df-z 9283 |
This theorem is referenced by: znnnlt1 9330 nn0n0n1ge2b 9361 eluzdc 9639 fzdcel 10069 fzn 10071 fzpreddisj 10100 fzp1disj 10109 fzneuz 10130 fznuz 10131 uznfz 10132 fzp1nel 10133 difelfznle 10164 fzodisj 10207 exfzdc 10269 modfzo0difsn 10425 fzfig 10460 iseqf1olemqk 10524 exp3val 10552 facdiv 10749 bcval5 10774 zfz1isolemiso 10850 2zsupmax 11265 2zinfmin 11282 summodclem3 11419 fprodntrivap 11623 alzdvds 11891 fzm1ndvds 11893 fzo0dvdseq 11894 n2dvds1 11948 dvdsbnd 11988 algcvgblem 12080 prmndvdsfaclt 12187 odzdvds 12276 pcprendvds 12321 pcdvdsb 12351 pc2dvds 12361 pcmpt 12374 pockthg 12388 prmunb 12393 1arith 12398 4sqlem11 12432 lgsdilem2 14890 uzdcinzz 15003 |
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