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Mirrors > Home > ILE Home > Th. List > lensymd | Unicode version |
Description: 'Less than or equal to' implies 'not less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
ltd.1 | |
ltd.2 | |
lensymd.3 |
Ref | Expression |
---|---|
lensymd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lensymd.3 | . 2 | |
2 | ltd.1 | . . 3 | |
3 | ltd.2 | . . 3 | |
4 | 2, 3 | lenltd 7880 | . 2 |
5 | 1, 4 | mpbid 146 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wcel 1480 class class class wbr 3929 cr 7619 clt 7800 cle 7801 |
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-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 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 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-ral 2421 df-rex 2422 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-br 3930 df-opab 3990 df-xp 4545 df-cnv 4547 df-xr 7804 df-le 7806 |
This theorem is referenced by: lbinf 8706 addmodlteq 10171 iseqf1olemab 10262 seq3f1olemqsumk 10272 seq3f1olemqsum 10273 zfz1isolemiso 10582 seq3coll 10585 maxleim 10977 maxabslemval 10980 cvgratz 11301 divalglemnqt 11617 bezoutlemsup 11697 dfgcd2 11702 lcmgcdlem 11758 trilpolemgt1 13232 trilpolemlt1 13234 |
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