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Mirrors > Home > ILE Home > Th. List > ltletrd | Unicode version |
Description: Transitive law deduction for 'less than', 'less than or equal to'. (Contributed by NM, 9-Jan-2006.) |
Ref | Expression |
---|---|
ltadd2d.1 | |
ltadd2d.2 | |
ltadd2d.3 | |
ltletrd.4 | |
ltletrd.5 |
Ref | Expression |
---|---|
ltletrd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltletrd.4 | . 2 | |
2 | ltletrd.5 | . 2 | |
3 | ltadd2d.1 | . . 3 | |
4 | ltadd2d.2 | . . 3 | |
5 | ltadd2d.3 | . . 3 | |
6 | ltletr 7967 | . . 3 | |
7 | 3, 4, 5, 6 | syl3anc 1220 | . 2 |
8 | 1, 2, 7 | mp2and 430 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 2128 class class class wbr 3966 cr 7732 clt 7913 cle 7914 |
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 4083 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-cnex 7824 ax-resscn 7825 ax-pre-ltwlin 7846 |
This theorem depends on definitions: df-bi 116 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-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-br 3967 df-opab 4027 df-xp 4593 df-cnv 4595 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-le 7919 |
This theorem is referenced by: lelttrdi 8302 lediv12a 8766 btwnapz 9295 rpgecl 9590 fznatpl1 9979 elfz1b 9993 exbtwnzlemstep 10151 ceiqle 10216 modqabs 10260 mulp1mod1 10268 seq3f1olemqsumk 10402 expgt1 10461 leexp2a 10476 bernneq3 10544 expnbnd 10545 nn0opthlem2d 10599 cvg1nlemres 10889 resqrexlemlo 10917 resqrexlemnmsq 10921 resqrexlemga 10927 abssubap0 10994 icodiamlt 11084 rpmaxcl 11127 reccn2ap 11214 divcnv 11398 cvgratnnlembern 11424 cvgratnnlemabsle 11428 fprodntrivap 11485 efcllemp 11559 sin01bnd 11658 cos01bnd 11659 sin01gt0 11662 cos12dec 11668 eirraplem 11677 dvdslelemd 11739 dvdsbnd 11844 znnen 12169 nninfdclemp1 12223 cnopnap 13036 dedekindeulemlu 13041 suplociccreex 13044 dedekindicclemlu 13050 dedekindicc 13053 ivthinclemlopn 13056 limcimolemlt 13075 limccnp2lem 13087 coseq00topi 13198 cosordlem 13212 logdivlti 13244 |
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