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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 7101 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)) | |
7 | 3, 4, 5, 6 | syl3anc 1135 | . 2 ⊢ (𝜑 → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)) |
8 | 1, 2, 7 | mp2and 409 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1393 class class class wbr 3764 ℝcr 6888 ≤ cle 7061 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-pre-ltwlin 6997 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-xp 4351 df-cnv 4353 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 |
This theorem is referenced by: eluzuzle 8481 fzdisj 8916 difelfzle 8992 flqwordi 9130 btwnzge0 9142 flqleceil 9159 bernneq 9369 bernneq3 9371 resqrexlemover 9608 resqrexlemdecn 9610 resqrexlemcalc3 9614 absle 9685 releabs 9692 climsqz 9855 climsqz2 9856 |
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