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Mirrors > Home > ILE Home > Th. List > lttri | GIF version |
Description: 'Less than' is transitive. Theorem I.17 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.) |
Ref | Expression |
---|---|
lt.1 | ⊢ 𝐴 ∈ ℝ |
lt.2 | ⊢ 𝐵 ∈ ℝ |
lt.3 | ⊢ 𝐶 ∈ ℝ |
Ref | Expression |
---|---|
lttri | ⊢ ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lt.1 | . 2 ⊢ 𝐴 ∈ ℝ | |
2 | lt.2 | . 2 ⊢ 𝐵 ∈ ℝ | |
3 | lt.3 | . 2 ⊢ 𝐶 ∈ ℝ | |
4 | lttr 7982 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) | |
5 | 1, 2, 3, 4 | mp3an 1332 | 1 ⊢ ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2141 class class class wbr 3987 ℝcr 7762 < clt 7943 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7854 ax-resscn 7855 ax-pre-lttrn 7877 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-xp 4615 df-pnf 7945 df-mnf 7946 df-ltxr 7948 |
This theorem is referenced by: 1lt3 9038 2lt4 9040 1lt4 9041 3lt5 9043 2lt5 9044 1lt5 9045 4lt6 9047 3lt6 9048 2lt6 9049 1lt6 9050 5lt7 9052 4lt7 9053 3lt7 9054 2lt7 9055 1lt7 9056 6lt8 9058 5lt8 9059 4lt8 9060 3lt8 9061 2lt8 9062 1lt8 9063 7lt9 9065 6lt9 9066 5lt9 9067 4lt9 9068 3lt9 9069 2lt9 9070 1lt9 9071 8lt10 9463 7lt10 9464 6lt10 9465 5lt10 9466 4lt10 9467 3lt10 9468 2lt10 9469 1lt10 9470 sincos2sgn 11717 cos12dec 11719 epos 11732 ene1 11736 eap1 11737 reeff1o 13449 pipos 13464 pigt3 13520 apdiff 14042 |
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