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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 8871 | . . . . . 6 ⊢ < Or ℝ* | |
2 | sowlin 4075 | . . . . . 6 ⊢ (( < Or ℝ* ∧ (𝐶 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) → (𝐶 < 𝐴 → (𝐶 < 𝐵 ∨ 𝐵 < 𝐴))) | |
3 | 1, 2 | mpan 414 | . . . . 5 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 < 𝐴 → (𝐶 < 𝐵 ∨ 𝐵 < 𝐴))) |
4 | 3 | 3coml 1145 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐶 < 𝐴 → (𝐶 < 𝐵 ∨ 𝐵 < 𝐴))) |
5 | orcom 679 | . . . 4 ⊢ ((𝐶 < 𝐵 ∨ 𝐵 < 𝐴) ↔ (𝐵 < 𝐴 ∨ 𝐶 < 𝐵)) | |
6 | 4, 5 | syl6ib 159 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐶 < 𝐴 → (𝐵 < 𝐴 ∨ 𝐶 < 𝐵))) |
7 | 6 | con3d 593 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (¬ (𝐵 < 𝐴 ∨ 𝐶 < 𝐵) → ¬ 𝐶 < 𝐴)) |
8 | xrlenlt 7177 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
9 | 8 | 3adant3 958 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) |
10 | xrlenlt 7177 | . . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐵 ≤ 𝐶 ↔ ¬ 𝐶 < 𝐵)) | |
11 | 10 | 3adant1 956 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐵 ≤ 𝐶 ↔ ¬ 𝐶 < 𝐵)) |
12 | 9, 11 | anbi12d 456 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐶 < 𝐵))) |
13 | ioran 701 | . . 3 ⊢ (¬ (𝐵 < 𝐴 ∨ 𝐶 < 𝐵) ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐶 < 𝐵)) | |
14 | 12, 13 | syl6bbr 196 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) ↔ ¬ (𝐵 < 𝐴 ∨ 𝐶 < 𝐵))) |
15 | xrlenlt 7177 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ 𝐶 ↔ ¬ 𝐶 < 𝐴)) | |
16 | 15 | 3adant2 957 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ 𝐶 ↔ ¬ 𝐶 < 𝐴)) |
17 | 7, 14, 16 | 3imtr4d 201 | 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 102 ↔ wb 103 ∨ wo 661 ∧ w3a 919 ∈ wcel 1433 class class class wbr 3785 Or wor 4050 ℝ*cxr 7152 < clt 7153 ≤ 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-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 |
This theorem depends on definitions: df-bi 115 df-3or 920 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-po 4051 df-iso 4052 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: xrletrd 8882 icc0r 8949 iccss 8964 icossico 8966 iccss2 8967 iccssico 8968 |
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