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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 8075 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) | |
7 | 3, 4, 5, 6 | syl3anc 1249 | . 2 ⊢ (𝜑 → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) |
8 | 1, 2, 7 | mp2and 433 | 1 ⊢ (𝜑 → 𝐴 < 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2160 class class class wbr 4018 ℝcr 7839 < clt 8021 ≤ cle 8022 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7931 ax-resscn 7932 ax-pre-ltwlin 7953 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-xp 4650 df-cnv 4652 df-pnf 8023 df-mnf 8024 df-xr 8025 df-ltxr 8026 df-le 8027 |
This theorem is referenced by: lt2msq1 8871 ledivp1 8889 suprzclex 9380 btwnapz 9412 ge0p1rp 9714 elfzolt3 10186 exbtwnz 10280 btwnzge0 10330 flltdivnn0lt 10334 modqid 10379 mulqaddmodid 10394 modqsubdir 10423 nn0opthlem2d 10732 bcp1nk 10773 zfz1isolemiso 10850 resqrexlemover 11050 resqrexlemnm 11058 resqrexlemcvg 11059 resqrexlemglsq 11062 resqrexlemga 11063 abslt 11128 abs3lem 11151 fzomaxdiflem 11152 icodiamlt 11220 maxltsup 11258 reccn2ap 11352 expcnvre 11542 absltap 11548 cvgratnnlemfm 11568 cvgratnnlemrate 11569 mertenslemi1 11574 ef01bndlem 11795 sin01bnd 11796 cos01bnd 11797 eirraplem 11815 dvdslelemd 11880 isprm5lem 12172 sqrt2irrap 12211 eulerthlemrprm 12260 eulerthlema 12261 pcfaclem 12380 pockthg 12388 4sqlem11 12432 4sqlem12 12433 4sqlem13m 12434 ssblex 14383 dedekindeulemuub 14547 dedekindeulemlu 14551 suplociccreex 14554 dedekindicclemuub 14556 dedekindicclemlu 14560 dedekindicc 14563 ivthinclemuopn 14568 dveflem 14639 coseq00topi 14708 coseq0negpitopi 14709 cosordlem 14722 logbgcd1irraplemexp 14838 lgsdirprm 14888 2sqlem8 14923 qdencn 15229 cvgcmp2nlemabs 15234 |
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