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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 8062 |
. 2
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5 | 1, 2, 3, 4 | mp3an 1348 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-pre-lttrn 7956 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-xp 4650 df-pnf 8025 df-mnf 8026 df-ltxr 8028 |
This theorem is referenced by: 1lt3 9121 2lt4 9123 1lt4 9124 3lt5 9126 2lt5 9127 1lt5 9128 4lt6 9130 3lt6 9131 2lt6 9132 1lt6 9133 5lt7 9135 4lt7 9136 3lt7 9137 2lt7 9138 1lt7 9139 6lt8 9141 5lt8 9142 4lt8 9143 3lt8 9144 2lt8 9145 1lt8 9146 7lt9 9148 6lt9 9149 5lt9 9150 4lt9 9151 3lt9 9152 2lt9 9153 1lt9 9154 8lt10 9546 7lt10 9547 6lt10 9548 5lt10 9549 4lt10 9550 3lt10 9551 2lt10 9552 1lt10 9553 sincos2sgn 11808 cos12dec 11810 epos 11823 ene1 11827 eap1 11828 reeff1o 14671 pipos 14686 pigt3 14742 apdiff 15275 |
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