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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 7859 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) | |
7 | 3, 4, 5, 6 | syl3anc 1216 | . 2 ⊢ (𝜑 → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) |
8 | 1, 2, 7 | mp2and 429 | 1 ⊢ (𝜑 → 𝐴 < 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1480 class class class wbr 3929 ℝcr 7626 < clt 7807 ≤ cle 7808 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7718 ax-resscn 7719 ax-pre-ltwlin 7740 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-cnv 4547 df-pnf 7809 df-mnf 7810 df-xr 7811 df-ltxr 7812 df-le 7813 |
This theorem is referenced by: lt2msq1 8650 ledivp1 8668 suprzclex 9156 btwnapz 9188 ge0p1rp 9480 elfzolt3 9941 exbtwnz 10035 btwnzge0 10080 flltdivnn0lt 10084 modqid 10129 mulqaddmodid 10144 modqsubdir 10173 nn0opthlem2d 10474 bcp1nk 10515 zfz1isolemiso 10589 resqrexlemover 10789 resqrexlemnm 10797 resqrexlemcvg 10798 resqrexlemglsq 10801 resqrexlemga 10802 abslt 10867 abs3lem 10890 fzomaxdiflem 10891 icodiamlt 10959 maxltsup 10997 reccn2ap 11089 expcnvre 11279 absltap 11285 cvgratnnlemfm 11305 cvgratnnlemrate 11306 mertenslemi1 11311 ef01bndlem 11470 sin01bnd 11471 cos01bnd 11472 eirraplem 11490 dvdslelemd 11548 sqrt2irrap 11865 ssblex 12610 dedekindeulemuub 12774 dedekindeulemlu 12778 suplociccreex 12781 dedekindicclemuub 12783 dedekindicclemlu 12787 dedekindicc 12790 ivthinclemuopn 12795 dveflem 12865 coseq00topi 12926 coseq0negpitopi 12927 cosordlem 12940 qdencn 13232 cvgcmp2nlemabs 13237 |
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