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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 |
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ltd.2 |
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ltled.1 |
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Ref | Expression |
---|---|
ltled |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltled.1 |
. 2
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2 | ltd.1 |
. . 3
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3 | ltd.2 |
. . 3
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4 | ltle 7635 |
. . 3
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5 | 2, 3, 4 | syl2anc 404 |
. 2
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6 | 1, 5 | mpd 13 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3965 ax-pow 4017 ax-pr 4047 ax-un 4271 ax-setind 4368 ax-cnex 7499 ax-resscn 7500 ax-pre-ltirr 7520 ax-pre-lttrn 7522 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-rab 2369 df-v 2624 df-dif 3004 df-un 3006 df-in 3008 df-ss 3015 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-br 3854 df-opab 3908 df-xp 4460 df-cnv 4462 df-pnf 7587 df-mnf 7588 df-xr 7589 df-ltxr 7590 df-le 7591 |
This theorem is referenced by: ltnsymd 7666 addgt0d 8061 lt2addd 8107 lt2msq1 8409 lediv12a 8418 ledivp1 8427 nn2ge 8518 fznatpl1 9553 exbtwnzlemex 9724 apbtwnz 9744 iseqf1olemkle 9976 expnbnd 10140 cvg1nlemres 10481 resqrexlemnm 10514 resqrexlemcvg 10515 resqrexlemglsq 10518 sqrtgt0 10530 leabs 10570 ltabs 10583 abslt 10584 absle 10585 maxabslemab 10702 2zsupmax 10720 fsum3cvg3 10852 divcnv 10954 expcnvre 10960 absltap 10966 cvgratnnlemnexp 10981 cvgratnnlemmn 10982 cvgratnnlemfm 10986 mertenslemi1 10992 dvdslelemd 11185 divalglemnn 11259 divalglemeuneg 11264 lcmgcdlem 11400 znege1 11497 sqrt2irraplemnn 11498 strleund 11645 |
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