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Mirrors > Home > ILE Home > Th. List > letrd | GIF version |
Description: Transitive law deduction for 'less than or equal to'. (Contributed by NM, 20-May-2005.) |
Ref | Expression |
---|---|
ltd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
letrd.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
letrd.4 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
letrd.5 | ⊢ (𝜑 → 𝐵 ≤ 𝐶) |
Ref | Expression |
---|---|
letrd | ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | letrd.4 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
2 | letrd.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 | letr 7194 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)) | |
7 | 3, 4, 5, 6 | syl3anc 1169 | . 2 ⊢ (𝜑 → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)) |
8 | 1, 2, 7 | mp2and 423 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∈ wcel 1433 class class class wbr 3785 ℝcr 6980 ≤ cle 7154 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-pre-ltwlin 7089 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-xp 4369 df-cnv 4371 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 |
This theorem is referenced by: eluzuzle 8627 fzdisj 9071 difelfzle 9145 flqwordi 9290 btwnzge0 9302 flqleceil 9319 modqltm1p1mod 9378 bernneq 9593 bernneq3 9595 nn0opthlem2d 9648 faclbnd 9668 facubnd 9672 resqrexlemover 9896 resqrexlemdecn 9898 resqrexlemcalc3 9902 absle 9975 releabs 9982 maxleastb 10100 climsqz 10173 climsqz2 10174 divalglemnqt 10320 infssuzex 10345 ncoprmgcdne1b 10471 |
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