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Mirrors > Home > ILE Home > Th. List > lelttrd | GIF version |
Description: Transitive law deduction for 'less than or equal to', 'less than'. (Contributed by NM, 8-Jan-2006.) |
Ref | Expression |
---|---|
ltd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
letrd.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
lelttrd.4 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
lelttrd.5 | ⊢ (𝜑 → 𝐵 < 𝐶) |
Ref | Expression |
---|---|
lelttrd | ⊢ (𝜑 → 𝐴 < 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lelttrd.4 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
2 | lelttrd.5 | . 2 ⊢ (𝜑 → 𝐵 < 𝐶) | |
3 | ltd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | ltd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
5 | letrd.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
6 | lelttr 7876 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) | |
7 | 3, 4, 5, 6 | syl3anc 1217 | . 2 ⊢ (𝜑 → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) |
8 | 1, 2, 7 | mp2and 430 | 1 ⊢ (𝜑 → 𝐴 < 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1481 class class class wbr 3937 ℝcr 7643 < clt 7824 ≤ cle 7825 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-pre-ltwlin 7757 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-xp 4553 df-cnv 4555 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 |
This theorem is referenced by: lt2msq1 8667 ledivp1 8685 suprzclex 9173 btwnapz 9205 ge0p1rp 9502 elfzolt3 9965 exbtwnz 10059 btwnzge0 10104 flltdivnn0lt 10108 modqid 10153 mulqaddmodid 10168 modqsubdir 10197 nn0opthlem2d 10499 bcp1nk 10540 zfz1isolemiso 10614 resqrexlemover 10814 resqrexlemnm 10822 resqrexlemcvg 10823 resqrexlemglsq 10826 resqrexlemga 10827 abslt 10892 abs3lem 10915 fzomaxdiflem 10916 icodiamlt 10984 maxltsup 11022 reccn2ap 11114 expcnvre 11304 absltap 11310 cvgratnnlemfm 11330 cvgratnnlemrate 11331 mertenslemi1 11336 ef01bndlem 11499 sin01bnd 11500 cos01bnd 11501 eirraplem 11519 dvdslelemd 11577 sqrt2irrap 11894 ssblex 12639 dedekindeulemuub 12803 dedekindeulemlu 12807 suplociccreex 12810 dedekindicclemuub 12812 dedekindicclemlu 12816 dedekindicc 12819 ivthinclemuopn 12824 dveflem 12895 coseq00topi 12964 coseq0negpitopi 12965 cosordlem 12978 logbgcd1irraplemexp 13093 qdencn 13397 cvgcmp2nlemabs 13402 |
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