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Mirrors > Home > ILE Home > Th. List > lttri | Unicode version |
Description: 'Less than' is transitive. Theorem I.17 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.) |
Ref | Expression |
---|---|
lt.1 | |
lt.2 | |
lt.3 |
Ref | Expression |
---|---|
lttri |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lt.1 | . 2 | |
2 | lt.2 | . 2 | |
3 | lt.3 | . 2 | |
4 | lttr 7806 | . 2 | |
5 | 1, 2, 3, 4 | mp3an 1300 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 1465 class class class wbr 3899 cr 7587 clt 7768 |
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-lttrn 7702 |
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-pnf 7770 df-mnf 7771 df-ltxr 7773 |
This theorem is referenced by: 1lt3 8859 2lt4 8861 1lt4 8862 3lt5 8864 2lt5 8865 1lt5 8866 4lt6 8868 3lt6 8869 2lt6 8870 1lt6 8871 5lt7 8873 4lt7 8874 3lt7 8875 2lt7 8876 1lt7 8877 6lt8 8879 5lt8 8880 4lt8 8881 3lt8 8882 2lt8 8883 1lt8 8884 7lt9 8886 6lt9 8887 5lt9 8888 4lt9 8889 3lt9 8890 2lt9 8891 1lt9 8892 8lt10 9281 7lt10 9282 6lt10 9283 5lt10 9284 4lt10 9285 3lt10 9286 2lt10 9287 1lt10 9288 sincos2sgn 11399 cos12dec 11401 epos 11414 ene1 11418 eap1 11419 pipos 12796 pigt3 12852 |
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