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Mirrors > Home > ILE Home > Th. List > lelttrd | Unicode version |
Description: Transitive law deduction for 'less than or equal to', 'less than'. (Contributed by NM, 8-Jan-2006.) |
Ref | Expression |
---|---|
ltd.1 | |
ltd.2 | |
letrd.3 | |
lelttrd.4 | |
lelttrd.5 |
Ref | Expression |
---|---|
lelttrd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lelttrd.4 | . 2 | |
2 | lelttrd.5 | . 2 | |
3 | ltd.1 | . . 3 | |
4 | ltd.2 | . . 3 | |
5 | letrd.3 | . . 3 | |
6 | lelttr 7820 | . . 3 | |
7 | 3, 4, 5, 6 | syl3anc 1201 | . 2 |
8 | 1, 2, 7 | mp2and 429 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 1465 class class class wbr 3899 cr 7587 clt 7768 cle 7769 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-pre-ltwlin 7701 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-xp 4515 df-cnv 4517 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 |
This theorem is referenced by: lt2msq1 8611 ledivp1 8629 suprzclex 9117 btwnapz 9149 ge0p1rp 9441 elfzolt3 9902 exbtwnz 9996 btwnzge0 10041 flltdivnn0lt 10045 modqid 10090 mulqaddmodid 10105 modqsubdir 10134 nn0opthlem2d 10435 bcp1nk 10476 zfz1isolemiso 10550 resqrexlemover 10750 resqrexlemnm 10758 resqrexlemcvg 10759 resqrexlemglsq 10762 resqrexlemga 10763 abslt 10828 abs3lem 10851 fzomaxdiflem 10852 icodiamlt 10920 maxltsup 10958 reccn2ap 11050 expcnvre 11240 absltap 11246 cvgratnnlemfm 11266 cvgratnnlemrate 11267 mertenslemi1 11272 ef01bndlem 11390 sin01bnd 11391 cos01bnd 11392 eirraplem 11410 dvdslelemd 11468 sqrt2irrap 11785 ssblex 12527 dedekindeulemuub 12691 dedekindeulemlu 12695 suplociccreex 12698 dedekindicclemuub 12700 dedekindicclemlu 12704 dedekindicc 12707 ivthinclemuopn 12712 dveflem 12782 coseq00topi 12843 coseq0negpitopi 12844 cosordlem 12857 qdencn 13149 cvgcmp2nlemabs 13154 |
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