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Mirrors > Home > ILE Home > Th. List > letrd | Unicode version |
Description: Transitive law deduction for 'less than or equal to'. (Contributed by NM, 20-May-2005.) |
Ref | Expression |
---|---|
ltd.1 | |
ltd.2 | |
letrd.3 | |
letrd.4 | |
letrd.5 |
Ref | Expression |
---|---|
letrd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | letrd.4 | . 2 | |
2 | letrd.5 | . 2 | |
3 | ltd.1 | . . 3 | |
4 | ltd.2 | . . 3 | |
5 | letrd.3 | . . 3 | |
6 | letr 7815 | . . 3 | |
7 | 3, 4, 5, 6 | syl3anc 1201 | . 2 |
8 | 1, 2, 7 | mp2and 429 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 1465 class class class wbr 3899 cr 7587 cle 7769 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-pre-ltwlin 7701 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-xp 4515 df-cnv 4517 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 |
This theorem is referenced by: eluzuzle 9302 fzdisj 9800 difelfzle 9879 flqwordi 10029 btwnzge0 10041 flqleceil 10058 modqltm1p1mod 10117 seq3split 10220 iseqf1olemqcl 10227 iseqf1olemnab 10229 iseqf1olemab 10230 seq3f1olemqsumkj 10239 seq3f1olemqsumk 10240 seq3f1olemqsum 10241 bernneq 10380 bernneq3 10382 nn0opthlem2d 10435 faclbnd 10455 facubnd 10459 seq3coll 10553 resqrexlemover 10750 resqrexlemdecn 10752 resqrexlemcalc3 10756 absle 10829 releabs 10836 maxleastb 10954 climsqz 11072 climsqz2 11073 fsum3cvg3 11133 expcnvap0 11239 geolim2 11249 cvgratnnlemabsle 11264 cvgratnnlemfm 11266 cvgratnnlemrate 11267 cvgratz 11269 mertenslem2 11273 eftlub 11323 cos12dec 11401 divalglemnqt 11544 infssuzex 11569 ncoprmgcdne1b 11697 ennnfoneleminc 11851 ennnfonelemkh 11852 strleund 11974 suplociccex 12699 ivthinclemlopn 12710 ivthinclemuopn 12712 dveflem 12782 cosordlem 12857 |
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