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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 7819 | . . 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 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-ltirr 7700 ax-pre-lttrn 7702 |
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: ltnsymd 7850 addgt0d 8251 lt2addd 8297 lt2msq1 8611 lediv12a 8620 ledivp1 8629 nn2ge 8721 fznatpl1 9824 exbtwnzlemex 9995 apbtwnz 10015 iseqf1olemkle 10225 expnbnd 10383 cvg1nlemres 10725 resqrexlemnm 10758 resqrexlemcvg 10759 resqrexlemglsq 10762 sqrtgt0 10774 leabs 10814 ltabs 10827 abslt 10828 absle 10829 maxabslemab 10946 2zsupmax 10965 xrmaxiflemab 10984 fsum3cvg3 11133 divcnv 11234 expcnvre 11240 absltap 11246 cvgratnnlemnexp 11261 cvgratnnlemmn 11262 cvgratnnlemfm 11266 mertenslemi1 11272 cos12dec 11401 dvdslelemd 11468 divalglemnn 11542 divalglemeuneg 11547 lcmgcdlem 11685 znege1 11783 sqrt2irraplemnn 11784 ennnfonelemex 11854 strleund 11974 suplociccreex 12698 ivthinclemlm 12708 ivthinclemum 12709 ivthinclemlopn 12710 ivthinclemuopn 12712 ivthdec 12718 dveflem 12782 sin0pilem1 12789 sin0pilem2 12790 coseq0negpitopi 12844 tangtx 12846 cosq34lt1 12858 cos02pilt1 12859 |
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