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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 10716 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 ≤ 𝐶 ↔ (𝐵 < 𝐶 ∨ 𝐵 = 𝐶))) | |
2 | 1 | 3adant1 1122 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 ≤ 𝐶 ↔ (𝐵 < 𝐶 ∨ 𝐵 = 𝐶))) |
3 | lttr 10706 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) | |
4 | 3 | expcomd 417 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 < 𝐶 → (𝐴 < 𝐵 → 𝐴 < 𝐶))) |
5 | breq2 5062 | . . . . . 6 ⊢ (𝐵 = 𝐶 → (𝐴 < 𝐵 ↔ 𝐴 < 𝐶)) | |
6 | 5 | biimpd 230 | . . . . 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 853 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐵 < 𝐶 ∨ 𝐵 = 𝐶) → (𝐴 < 𝐵 → 𝐴 < 𝐶))) |
9 | 2, 8 | sylbid 241 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 ≤ 𝐶 → (𝐴 < 𝐵 → 𝐴 < 𝐶))) |
10 | 9 | impcomd 412 | 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: ltleletr 10722 ltletri 10757 ltletrd 10789 ltleadd 11112 lediv12a 11522 nngt0 11657 nnrecgt0 11669 elnnnn0c 11931 elnnz1 11997 zltp1le 12021 uz3m2nn 12280 zbtwnre 12335 ledivge1le 12450 addlelt 12493 qbtwnre 12582 xlemul1a 12671 xrsupsslem 12690 zltaddlt1le 12880 elfzodifsumelfzo 13093 ssfzo12bi 13122 elfznelfzo 13132 ceile 13207 swrdswrd 14057 swrdccatin1 14077 repswswrd 14136 sqrlem4 14595 resqrex 14600 caubnd 14708 rlim2lt 14844 cos01gt0 15534 ruclem12 15584 oddge22np1 15688 sadcaddlem 15796 nn0seqcvgd 15904 coprm 16045 prmgaplem7 16383 prmlem1 16431 prmlem2 16443 icoopnst 23472 ovollb2lem 24018 dvcnvrelem1 24543 aaliou 24856 tanord 25049 logdivlti 25130 logdivlt 25131 ftalem2 25579 gausslemma2dlem1a 25869 pntlem3 26113 crctcshwlkn0lem3 27518 nn0prpwlem 33568 isbasisrelowllem1 34519 isbasisrelowllem2 34520 ltflcei 34762 tan2h 34766 poimirlem29 34803 poimirlem32 34806 stoweidlem26 42192 stoweid 42229 2leaddle2 43379 gbegt5 43773 gbowgt5 43774 sgoldbeven3prm 43795 nnsum4primesodd 43808 nnsum4primesoddALTV 43809 evengpoap3 43811 bgoldbnnsum3prm 43816 cznnring 44125 nn0sumltlt 44296 rege1logbrege0 44516 rege1logbzge0 44517 fllog2 44526 dignn0ldlem 44560 |
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