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Mirrors > Home > MPE Home > Th. List > lttri | Structured version Visualization version 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 10114 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) | |
5 | 1, 2, 3, 4 | mp3an 1424 | 1 ⊢ ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∈ wcel 1990 class class class wbr 4653 ℝcr 9935 < clt 10074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-pre-lttrn 10011 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 |
This theorem is referenced by: 1lt3 11196 2lt4 11198 1lt4 11199 3lt5 11201 2lt5 11202 1lt5 11203 4lt6 11205 3lt6 11206 2lt6 11207 1lt6 11208 5lt7 11210 4lt7 11211 3lt7 11212 2lt7 11213 1lt7 11214 6lt8 11216 5lt8 11217 4lt8 11218 3lt8 11219 2lt8 11220 1lt8 11221 7lt9 11223 6lt9 11224 5lt9 11225 4lt9 11226 3lt9 11227 2lt9 11228 1lt9 11229 8lt10OLD 11231 7lt10OLD 11232 6lt10OLD 11233 5lt10OLD 11234 4lt10OLD 11235 3lt10OLD 11236 2lt10OLD 11237 1lt10OLD 11238 8lt10 11674 7lt10 11675 6lt10 11676 5lt10 11677 4lt10 11678 3lt10 11679 2lt10 11680 1lt10 11681 sincos2sgn 14924 epos 14935 ene1 14938 dvdslelem 15031 oppcbas 16378 sralem 19177 zlmlem 19865 psgnodpmr 19936 tnglem 22444 xrhmph 22746 vitalilem4 23380 pipos 24212 logneg 24334 asin1 24621 reasinsin 24623 atan1 24655 log2le1 24677 bposlem8 25016 bposlem9 25017 chebbnd1lem2 25159 chebbnd1lem3 25160 chebbnd1 25161 mulog2sumlem2 25224 pntibndlem1 25278 pntlemb 25286 pntlemk 25295 ttglem 25756 cchhllem 25767 axlowdimlem16 25837 dp2ltc 29594 sgnnbi 30607 sgnpbi 30608 signswch 30638 hgt750lem 30729 hgt750lem2 30730 logi 31620 cnndvlem1 32528 bj-minftyccb 33112 bj-pinftynminfty 33114 asindmre 33495 fdc 33541 fourierdlem94 40417 fourierdlem102 40425 fourierdlem103 40426 fourierdlem104 40427 fourierdlem112 40435 fourierdlem113 40436 fourierdlem114 40437 fouriersw 40448 etransclem23 40474 |
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