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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 8048 |
. . 3
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5 | 2, 3, 4 | syl2anc 411 |
. 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-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7905 ax-resscn 7906 ax-pre-ltirr 7926 ax-pre-lttrn 7928 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-xp 4634 df-cnv 4636 df-pnf 7997 df-mnf 7998 df-xr 7999 df-ltxr 8000 df-le 8001 |
This theorem is referenced by: ltnsymd 8080 addgt0d 8481 lt2addd 8527 lt2msq1 8845 lediv12a 8854 ledivp1 8863 nn2ge 8955 fznatpl1 10079 exbtwnzlemex 10253 apbtwnz 10277 iseqf1olemkle 10487 expnbnd 10647 nn0ltexp2 10692 cvg1nlemres 10997 resqrexlemnm 11030 resqrexlemcvg 11031 resqrexlemglsq 11034 sqrtgt0 11046 leabs 11086 ltabs 11099 abslt 11100 absle 11101 maxabslemab 11218 2zsupmax 11237 2zinfmin 11254 xrmaxiflemab 11258 fsum3cvg3 11407 divcnv 11508 expcnvre 11514 absltap 11520 cvgratnnlemnexp 11535 cvgratnnlemmn 11536 cvgratnnlemfm 11540 mertenslemi1 11546 cos12dec 11778 dvdslelemd 11852 divalglemnn 11926 divalglemeuneg 11931 lcmgcdlem 12080 isprm5lem 12144 znege1 12181 sqrt2irraplemnn 12182 eulerthlemrprm 12232 eulerthlema 12233 4sqlem7 12385 ennnfonelemex 12418 strleund 12565 suplociccreex 14242 ivthinclemlm 14252 ivthinclemum 14253 ivthinclemlopn 14254 ivthinclemuopn 14256 ivthdec 14262 dveflem 14327 efltlemlt 14335 sin0pilem1 14342 sin0pilem2 14343 coseq0negpitopi 14397 tangtx 14399 cosq34lt1 14411 cos02pilt1 14412 lgseisenlem1 14590 apdifflemf 14934 |
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