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Mirrors > Home > MPE Home > Th. List > lelttr | Structured version Visualization version GIF version |
Description: Transitive law. (Contributed by NM, 23-May-1999.) |
Ref | Expression |
---|---|
lelttr | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leloe 10716 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) | |
2 | 1 | 3adant3 1124 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) |
3 | lttr 10706 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) | |
4 | 3 | expd 416 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐵 < 𝐶 → 𝐴 < 𝐶))) |
5 | breq1 5061 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 < 𝐶 ↔ 𝐵 < 𝐶)) | |
6 | 5 | biimprd 249 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝐵 < 𝐶 → 𝐴 < 𝐶)) |
7 | 6 | a1i 11 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 = 𝐵 → (𝐵 < 𝐶 → 𝐴 < 𝐶))) |
8 | 4, 7 | jaod 853 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∨ 𝐴 = 𝐵) → (𝐵 < 𝐶 → 𝐴 < 𝐶))) |
9 | 2, 8 | sylbid 241 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 → (𝐵 < 𝐶 → 𝐴 < 𝐶))) |
10 | 9 | impd 411 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 841 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 class class class wbr 5058 ℝcr 10525 < clt 10664 ≤ cle 10665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 |
This theorem is referenced by: letr 10723 lelttri 10756 lelttrd 10787 letrp1 11473 ltmul12a 11485 ledivp1 11531 supmul1 11599 bndndx 11885 uzind 12063 fnn0ind 12070 rpnnen1lem5 12370 xrinfmsslem 12691 elfzo0z 13069 nn0p1elfzo 13070 fzofzim 13074 elfzodifsumelfzo 13093 flge 13165 flflp1 13167 flltdivnn0lt 13193 modfzo0difsn 13301 fsequb 13333 expnlbnd2 13585 ccat2s1fvw 13988 ccat2s1fvwOLD 13989 swrdswrd 14057 pfxccatin12lem3 14084 repswswrd 14136 caubnd2 14707 caubnd 14708 mulcn2 14942 cn1lem 14944 rlimo1 14963 o1rlimmul 14965 climsqz 14987 climsqz2 14988 rlimsqzlem 14995 climsup 15016 caucvgrlem2 15021 iseralt 15031 cvgcmp 15161 cvgcmpce 15163 ruclem3 15576 ruclem12 15584 ltoddhalfle 15700 algcvgblem 15911 ncoprmlnprm 16058 pclem 16165 infpn2 16239 gsummoncoe1 20402 mp2pm2mplem4 21347 metss2lem 23050 ngptgp 23174 nghmcn 23283 iocopnst 23473 ovollb2lem 24018 ovolicc2lem4 24050 volcn 24136 ismbf3d 24184 dvcnvrelem1 24543 dvfsumrlim 24557 ulmcn 24916 mtest 24921 logdivlti 25130 isosctrlem1 25323 ftalem2 25579 chtub 25716 bposlem6 25793 gausslemma2dlem2 25871 chtppilim 25979 dchrisumlem3 25995 pntlem3 26113 clwlkclwwlklem2a 27704 vacn 28399 nmcvcn 28400 blocni 28510 chscllem2 29343 lnconi 29738 staddi 29951 stadd3i 29953 ltflcei 34762 poimirlem29 34803 geomcau 34917 heibor1lem 34970 bfplem2 34984 rrncmslem 34993 climinf 41767 leltletr 43374 zm1nn 43383 iccpartigtl 43430 tgoldbach 43829 ply1mulgsumlem2 44339 difmodm1lt 44480 |
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