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Mirrors > Home > ILE Home > Th. List > lttrd | Unicode version |
Description: Transitive law deduction for 'less than'. (Contributed by NM, 9-Jan-2006.) |
Ref | Expression |
---|---|
ltd.1 | |
ltd.2 | |
letrd.3 | |
lttrd.4 | |
lttrd.5 |
Ref | Expression |
---|---|
lttrd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lttrd.4 | . 2 | |
2 | lttrd.5 | . 2 | |
3 | ltd.1 | . . 3 | |
4 | ltd.2 | . . 3 | |
5 | letrd.3 | . . 3 | |
6 | lttr 7838 | . . 3 | |
7 | 3, 4, 5, 6 | syl3anc 1216 | . 2 |
8 | 1, 2, 7 | mp2and 429 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 1480 class class class wbr 3929 cr 7619 clt 7800 |
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-13 1491 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 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-pre-lttrn 7734 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 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-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 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-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-pnf 7802 df-mnf 7803 df-ltxr 7805 |
This theorem is referenced by: exbtwnzlemex 10027 rebtwn2z 10032 qbtwnrelemcalc 10033 expgt1 10331 ltexp2a 10345 expnlbnd2 10417 expcanlem 10462 expcan 10463 cvg1nlemcxze 10754 cvg1nlemcau 10756 cvg1nlemres 10757 recvguniqlem 10766 resqrexlemdecn 10784 resqrexlemcvg 10791 resqrexlemga 10795 qdenre 10974 reccn2ap 11082 georeclim 11282 geoisumr 11287 cvgratz 11301 efcllemp 11364 efgt1 11403 cos12dec 11474 dvdslelemd 11541 ivthinclemlr 12784 ivthinclemur 12786 limcimolemlt 12802 sin0pilem1 12862 pilem3 12864 coseq0negpitopi 12917 tangtx 12919 cos02pilt1 12932 cvgcmp2nlemabs 13227 trilpolemlt1 13234 |
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