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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 11304 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) | |
2 | 1 | 3adant3 1132 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) |
3 | lttr 11294 | . . . . 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 5151 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 < 𝐶 ↔ 𝐵 < 𝐶)) | |
6 | 5 | biimprd 247 | . . . . 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 857 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∨ 𝐴 = 𝐵) → (𝐵 < 𝐶 → 𝐴 < 𝐶))) |
9 | 2, 8 | sylbid 239 | . 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 205 ∧ wa 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 class class class wbr 5148 ℝcr 11111 < clt 11252 ≤ cle 11253 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-resscn 11169 ax-pre-lttri 11186 ax-pre-lttrn 11187 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 |
This theorem is referenced by: leltletr 11309 letr 11312 lelttri 11345 lelttrd 11376 letrp1 12062 ltmul12a 12074 ledivp1 12120 supmul1 12187 bndndx 12475 uzind 12658 fnn0ind 12665 rpnnen1lem5 12969 xrinfmsslem 13291 elfzo0z 13678 nn0p1elfzo 13679 fzofzim 13683 elfzodifsumelfzo 13702 flge 13774 flflp1 13776 flltdivnn0lt 13802 modfzo0difsn 13912 fsequb 13944 expnlbnd2 14201 ccat2s1fvw 14592 swrdswrd 14659 pfxccatin12lem3 14686 repswswrd 14738 caubnd2 15308 caubnd 15309 mulcn2 15544 cn1lem 15546 rlimo1 15565 o1rlimmul 15567 climsqz 15589 climsqz2 15590 rlimsqzlem 15599 climsup 15620 caucvgrlem2 15625 iseralt 15635 cvgcmp 15766 cvgcmpce 15768 ruclem3 16180 ruclem12 16188 ltoddhalfle 16308 algcvgblem 16518 ncoprmlnprm 16668 pclem 16775 infpn2 16850 gsummoncoe1 22048 mp2pm2mplem4 22531 metss2lem 24240 ngptgp 24365 nghmcn 24482 iocopnst 24680 ovollb2lem 25229 ovolicc2lem4 25261 volcn 25347 ismbf3d 25395 dvcnvrelem1 25758 dvfsumrlim 25772 ulmcn 26135 mtest 26140 logdivlti 26352 isosctrlem1 26547 ftalem2 26802 chtub 26939 bposlem6 27016 gausslemma2dlem2 27094 chtppilim 27202 dchrisumlem3 27218 pntlem3 27336 clwlkclwwlklem2a 29506 vacn 30202 nmcvcn 30203 blocni 30313 chscllem2 31146 lnconi 31541 staddi 31754 stadd3i 31756 ltflcei 36779 poimirlem29 36820 geomcau 36930 heibor1lem 36980 bfplem2 36994 rrncmslem 37003 climinf 44621 zm1nn 46309 iccpartigtl 46390 tgoldbach 46784 ply1mulgsumlem2 47156 difmodm1lt 47296 |
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