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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 |
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lt.2 |
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lt.3 |
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Ref | Expression |
---|---|
lttri |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lt.1 |
. 2
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2 | lt.2 |
. 2
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3 | lt.3 |
. 2
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4 | lttr 7761 |
. 2
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5 | 1, 2, 3, 4 | mp3an 1298 |
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 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-cnex 7636 ax-resscn 7637 ax-pre-lttrn 7659 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-nel 2378 df-ral 2395 df-rex 2396 df-rab 2399 df-v 2659 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-opab 3950 df-xp 4505 df-pnf 7726 df-mnf 7727 df-ltxr 7729 |
This theorem is referenced by: 1lt3 8795 2lt4 8797 1lt4 8798 3lt5 8800 2lt5 8801 1lt5 8802 4lt6 8804 3lt6 8805 2lt6 8806 1lt6 8807 5lt7 8809 4lt7 8810 3lt7 8811 2lt7 8812 1lt7 8813 6lt8 8815 5lt8 8816 4lt8 8817 3lt8 8818 2lt8 8819 1lt8 8820 7lt9 8822 6lt9 8823 5lt9 8824 4lt9 8825 3lt9 8826 2lt9 8827 1lt9 8828 8lt10 9217 7lt10 9218 6lt10 9219 5lt10 9220 4lt10 9221 3lt10 9222 2lt10 9223 1lt10 9224 sincos2sgn 11323 epos 11335 ene1 11339 eap1 11340 |
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