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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 7854 | . . 3 | |
5 | 2, 3, 4 | syl2anc 408 | . 2 |
6 | 1, 5 | mpd 13 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 1480 class class class wbr 3929 cr 7622 clt 7803 cle 7804 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7714 ax-resscn 7715 ax-pre-ltirr 7735 ax-pre-lttrn 7737 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-cnv 4547 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 |
This theorem is referenced by: ltnsymd 7885 addgt0d 8286 lt2addd 8332 lt2msq1 8646 lediv12a 8655 ledivp1 8664 nn2ge 8756 fznatpl1 9859 exbtwnzlemex 10030 apbtwnz 10050 iseqf1olemkle 10260 expnbnd 10418 cvg1nlemres 10760 resqrexlemnm 10793 resqrexlemcvg 10794 resqrexlemglsq 10797 sqrtgt0 10809 leabs 10849 ltabs 10862 abslt 10863 absle 10864 maxabslemab 10981 2zsupmax 11000 xrmaxiflemab 11019 fsum3cvg3 11168 divcnv 11269 expcnvre 11275 absltap 11281 cvgratnnlemnexp 11296 cvgratnnlemmn 11297 cvgratnnlemfm 11301 mertenslemi1 11307 cos12dec 11477 dvdslelemd 11544 divalglemnn 11618 divalglemeuneg 11623 lcmgcdlem 11761 znege1 11859 sqrt2irraplemnn 11860 ennnfonelemex 11930 strleund 12050 suplociccreex 12774 ivthinclemlm 12784 ivthinclemum 12785 ivthinclemlopn 12786 ivthinclemuopn 12788 ivthdec 12794 dveflem 12858 sin0pilem1 12865 sin0pilem2 12866 coseq0negpitopi 12920 tangtx 12922 cosq34lt1 12934 cos02pilt1 12935 |
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