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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 7669 |
. . 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 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 ax-pre-ltirr 7554 ax-pre-lttrn 7556 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-rab 2379 df-v 2635 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-xp 4473 df-cnv 4475 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 |
This theorem is referenced by: ltnsymd 7700 addgt0d 8095 lt2addd 8141 lt2msq1 8443 lediv12a 8452 ledivp1 8461 nn2ge 8553 fznatpl1 9639 exbtwnzlemex 9810 apbtwnz 9830 iseqf1olemkle 10034 expnbnd 10192 cvg1nlemres 10533 resqrexlemnm 10566 resqrexlemcvg 10567 resqrexlemglsq 10570 sqrtgt0 10582 leabs 10622 ltabs 10635 abslt 10636 absle 10637 maxabslemab 10754 2zsupmax 10772 xrmaxiflemab 10790 fsum3cvg3 10939 divcnv 11040 expcnvre 11046 absltap 11052 cvgratnnlemnexp 11067 cvgratnnlemmn 11068 cvgratnnlemfm 11072 mertenslemi1 11078 dvdslelemd 11271 divalglemnn 11345 divalglemeuneg 11350 lcmgcdlem 11486 znege1 11583 sqrt2irraplemnn 11584 strleund 11731 |
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