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Mirrors > Home > ILE Home > Th. List > xrletr | GIF version |
Description: Transitive law for ordering on extended reals. (Contributed by NM, 9-Feb-2006.) |
Ref | Expression |
---|---|
xrletr | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrltso 9612 | . . . . . 6 ⊢ < Or ℝ* | |
2 | sowlin 4250 | . . . . . 6 ⊢ (( < Or ℝ* ∧ (𝐶 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) → (𝐶 < 𝐴 → (𝐶 < 𝐵 ∨ 𝐵 < 𝐴))) | |
3 | 1, 2 | mpan 421 | . . . . 5 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 < 𝐴 → (𝐶 < 𝐵 ∨ 𝐵 < 𝐴))) |
4 | 3 | 3coml 1189 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐶 < 𝐴 → (𝐶 < 𝐵 ∨ 𝐵 < 𝐴))) |
5 | orcom 718 | . . . 4 ⊢ ((𝐶 < 𝐵 ∨ 𝐵 < 𝐴) ↔ (𝐵 < 𝐴 ∨ 𝐶 < 𝐵)) | |
6 | 4, 5 | syl6ib 160 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐶 < 𝐴 → (𝐵 < 𝐴 ∨ 𝐶 < 𝐵))) |
7 | 6 | con3d 621 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (¬ (𝐵 < 𝐴 ∨ 𝐶 < 𝐵) → ¬ 𝐶 < 𝐴)) |
8 | xrlenlt 7853 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
9 | 8 | 3adant3 1002 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) |
10 | xrlenlt 7853 | . . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐵 ≤ 𝐶 ↔ ¬ 𝐶 < 𝐵)) | |
11 | 10 | 3adant1 1000 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐵 ≤ 𝐶 ↔ ¬ 𝐶 < 𝐵)) |
12 | 9, 11 | anbi12d 465 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐶 < 𝐵))) |
13 | ioran 742 | . . 3 ⊢ (¬ (𝐵 < 𝐴 ∨ 𝐶 < 𝐵) ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐶 < 𝐵)) | |
14 | 12, 13 | syl6bbr 197 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) ↔ ¬ (𝐵 < 𝐴 ∨ 𝐶 < 𝐵))) |
15 | xrlenlt 7853 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ 𝐶 ↔ ¬ 𝐶 < 𝐴)) | |
16 | 15 | 3adant2 1001 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ 𝐶 ↔ ¬ 𝐶 < 𝐴)) |
17 | 7, 14, 16 | 3imtr4d 202 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 698 ∧ w3a 963 ∈ wcel 1481 class class class wbr 3937 Or wor 4225 ℝ*cxr 7823 < 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-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 |
This theorem depends on definitions: df-bi 116 df-3or 964 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-po 4226 df-iso 4227 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: xrletrd 9625 xle2add 9692 icc0r 9739 iccss 9754 icossico 9756 iccss2 9757 iccssico 9758 bdxmet 12709 |
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