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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 10729 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) | |
2 | 1 | 3adant3 1128 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) |
3 | lttr 10719 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) | |
4 | 3 | expd 418 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐵 < 𝐶 → 𝐴 < 𝐶))) |
5 | breq1 5071 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 < 𝐶 ↔ 𝐵 < 𝐶)) | |
6 | 5 | biimprd 250 | . . . . 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 855 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∨ 𝐴 = 𝐵) → (𝐵 < 𝐶 → 𝐴 < 𝐶))) |
9 | 2, 8 | sylbid 242 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 → (𝐵 < 𝐶 → 𝐴 < 𝐶))) |
10 | 9 | impd 413 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ℝcr 10538 < clt 10677 ≤ cle 10678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-pre-lttri 10613 ax-pre-lttrn 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 |
This theorem is referenced by: letr 10736 lelttri 10769 lelttrd 10800 letrp1 11486 ltmul12a 11498 ledivp1 11544 supmul1 11612 bndndx 11899 uzind 12077 fnn0ind 12084 rpnnen1lem5 12383 xrinfmsslem 12704 elfzo0z 13082 nn0p1elfzo 13083 fzofzim 13087 elfzodifsumelfzo 13106 flge 13178 flflp1 13180 flltdivnn0lt 13206 modfzo0difsn 13314 fsequb 13346 expnlbnd2 13598 ccat2s1fvw 14000 ccat2s1fvwOLD 14001 swrdswrd 14069 pfxccatin12lem3 14096 repswswrd 14148 caubnd2 14719 caubnd 14720 mulcn2 14954 cn1lem 14956 rlimo1 14975 o1rlimmul 14977 climsqz 14999 climsqz2 15000 rlimsqzlem 15007 climsup 15028 caucvgrlem2 15033 iseralt 15043 cvgcmp 15173 cvgcmpce 15175 ruclem3 15588 ruclem12 15596 ltoddhalfle 15712 algcvgblem 15923 ncoprmlnprm 16070 pclem 16177 infpn2 16251 gsummoncoe1 20474 mp2pm2mplem4 21419 metss2lem 23123 ngptgp 23247 nghmcn 23356 iocopnst 23546 ovollb2lem 24091 ovolicc2lem4 24123 volcn 24209 ismbf3d 24257 dvcnvrelem1 24616 dvfsumrlim 24630 ulmcn 24989 mtest 24994 logdivlti 25205 isosctrlem1 25398 ftalem2 25653 chtub 25790 bposlem6 25867 gausslemma2dlem2 25945 chtppilim 26053 dchrisumlem3 26069 pntlem3 26187 clwlkclwwlklem2a 27778 vacn 28473 nmcvcn 28474 blocni 28584 chscllem2 29417 lnconi 29812 staddi 30025 stadd3i 30027 ltflcei 34882 poimirlem29 34923 geomcau 35036 heibor1lem 35089 bfplem2 35103 rrncmslem 35112 climinf 41894 leltletr 43500 zm1nn 43509 iccpartigtl 43590 tgoldbach 43989 ply1mulgsumlem2 44448 difmodm1lt 44589 |
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