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Mirrors > Home > ILE Home > Th. List > ltle | Unicode version |
Description: 'Less than' implies 'less than or equal to'. (Contributed by NM, 25-Aug-1999.) |
Ref | Expression |
---|---|
ltle |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltnsym 7963 | . 2 | |
2 | lenlt 7953 | . 2 | |
3 | 1, 2 | sylibrd 168 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wcel 2128 class class class wbr 3965 cr 7731 clt 7912 cle 7913 |
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 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-pre-ltirr 7844 ax-pre-lttrn 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 3773 df-br 3966 df-opab 4026 df-xp 4592 df-cnv 4594 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 |
This theorem is referenced by: ltlei 7978 ltled 7994 ltleap 8507 lep1 8716 lem1 8718 letrp1 8719 ltmul12a 8731 bndndx 9089 nn0ge0 9115 zletric 9211 zlelttric 9212 zltnle 9213 zleloe 9214 zdcle 9240 uzind 9275 fnn0ind 9280 eluz2b2 9514 rpge0 9573 zltaddlt1le 9911 difelfznle 10034 elfzouz2 10060 elfzo0le 10084 fzosplitprm1 10133 fzostep1 10136 qletric 10143 qlelttric 10144 qltnle 10145 expgt1 10457 expnlbnd2 10543 faclbnd 10615 caucvgrelemcau 10880 resqrexlemdecn 10912 mulcn2 11209 efcllemp 11555 sin01bnd 11654 cos01bnd 11655 sin01gt0 11658 cos01gt0 11659 absef 11666 efieq1re 11668 nn0o 11797 sincosq1lem 13146 tangtx 13159 |
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