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Mirrors > Home > ILE Home > Th. List > ltled | Unicode version |
Description: 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
ltd.1 | |
ltd.2 | |
ltled.1 |
Ref | Expression |
---|---|
ltled |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltled.1 | . 2 | |
2 | ltd.1 | . . 3 | |
3 | ltd.2 | . . 3 | |
4 | ltle 7964 | . . 3 | |
5 | 2, 3, 4 | syl2anc 409 | . 2 |
6 | 1, 5 | mpd 13 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 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: ltnsymd 7995 addgt0d 8396 lt2addd 8442 lt2msq1 8756 lediv12a 8765 ledivp1 8774 nn2ge 8866 fznatpl1 9978 exbtwnzlemex 10149 apbtwnz 10173 iseqf1olemkle 10383 expnbnd 10541 cvg1nlemres 10885 resqrexlemnm 10918 resqrexlemcvg 10919 resqrexlemglsq 10922 sqrtgt0 10934 leabs 10974 ltabs 10987 abslt 10988 absle 10989 maxabslemab 11106 2zsupmax 11125 xrmaxiflemab 11144 fsum3cvg3 11293 divcnv 11394 expcnvre 11400 absltap 11406 cvgratnnlemnexp 11421 cvgratnnlemmn 11422 cvgratnnlemfm 11426 mertenslemi1 11432 cos12dec 11664 dvdslelemd 11734 divalglemnn 11808 divalglemeuneg 11813 lcmgcdlem 11953 znege1 12052 sqrt2irraplemnn 12053 eulerthlemrprm 12103 eulerthlema 12104 ennnfonelemex 12143 strleund 12278 suplociccreex 13002 ivthinclemlm 13012 ivthinclemum 13013 ivthinclemlopn 13014 ivthinclemuopn 13016 ivthdec 13022 dveflem 13087 efltlemlt 13095 sin0pilem1 13102 sin0pilem2 13103 coseq0negpitopi 13157 tangtx 13159 cosq34lt1 13171 cos02pilt1 13172 apdifflemf 13617 |
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