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| Mirrors > Home > MPE Home > Th. List > ltletr | Structured version Visualization version GIF version | ||
| Description: Transitive law. (Contributed by NM, 25-Aug-1999.) |
| Ref | Expression |
|---|---|
| ltletr | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | leloe 11347 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 ≤ 𝐶 ↔ (𝐵 < 𝐶 ∨ 𝐵 = 𝐶))) | |
| 2 | 1 | 3adant1 1131 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 ≤ 𝐶 ↔ (𝐵 < 𝐶 ∨ 𝐵 = 𝐶))) |
| 3 | lttr 11337 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) | |
| 4 | 3 | expcomd 416 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 < 𝐶 → (𝐴 < 𝐵 → 𝐴 < 𝐶))) |
| 5 | breq2 5147 | . . . . . 6 ⊢ (𝐵 = 𝐶 → (𝐴 < 𝐵 ↔ 𝐴 < 𝐶)) | |
| 6 | 5 | biimpd 229 | . . . . 5 ⊢ (𝐵 = 𝐶 → (𝐴 < 𝐵 → 𝐴 < 𝐶)) |
| 7 | 6 | a1i 11 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 = 𝐶 → (𝐴 < 𝐵 → 𝐴 < 𝐶))) |
| 8 | 4, 7 | jaod 860 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐵 < 𝐶 ∨ 𝐵 = 𝐶) → (𝐴 < 𝐵 → 𝐴 < 𝐶))) |
| 9 | 2, 8 | sylbid 240 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 ≤ 𝐶 → (𝐴 < 𝐵 → 𝐴 < 𝐶))) |
| 10 | 9 | impcomd 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 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ℝcr 11154 < clt 11295 ≤ cle 11296 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 |
| This theorem is referenced by: ltleletr 11354 ltletri 11389 ltletrd 11421 ltleadd 11746 lediv12a 12161 nngt0 12297 nnrecgt0 12309 elnnnn0c 12571 elnnz1 12643 zltp1le 12667 uz3m2nn 12933 zbtwnre 12988 ledivge1le 13106 addlelt 13149 qbtwnre 13241 xlemul1a 13330 xrsupsslem 13349 zltaddlt1le 13545 elfzodifsumelfzo 13770 ssfzo12bi 13800 elfznelfzo 13811 ceile 13889 swrdswrd 14743 swrdccatin1 14763 repswswrd 14822 01sqrexlem4 15284 resqrex 15289 caubnd 15397 rlim2lt 15533 cos01gt0 16227 ruclem12 16277 oddge22np1 16386 sadcaddlem 16494 nn0seqcvgd 16607 coprm 16748 prmgaplem7 17095 prmlem1 17145 prmlem2 17157 icoopnst 24969 ovollb2lem 25523 dvcnvrelem1 26056 aaliou 26380 tanord 26580 logdivlti 26662 logdivlt 26663 ftalem2 27117 gausslemma2dlem1a 27409 pntlem3 27653 crctcshwlkn0lem3 29832 nn0prpwlem 36323 isbasisrelowllem1 37356 isbasisrelowllem2 37357 ltflcei 37615 tan2h 37619 poimirlem29 37656 poimirlem32 37659 2xp3dxp2ge1d 42242 stoweidlem26 46041 stoweid 46078 2leaddle2 47310 gbegt5 47748 gbowgt5 47749 sgoldbeven3prm 47770 nnsum4primesodd 47783 nnsum4primesoddALTV 47784 evengpoap3 47786 bgoldbnnsum3prm 47791 cznnring 48178 nn0sumltlt 48266 rege1logbrege0 48479 rege1logbzge0 48480 fllog2 48489 dignn0ldlem 48523 |
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