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Mirrors > Home > ILE Home > Th. List > xrlelttr | GIF version |
Description: Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.) |
Ref | Expression |
---|---|
xrlelttr | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprl 521 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶)) → 𝐴 ≤ 𝐵) | |
2 | simpl1 989 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶)) → 𝐴 ∈ ℝ*) | |
3 | simpl2 990 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ ℝ*) | |
4 | xrlenlt 7954 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
5 | 2, 3, 4 | syl2anc 409 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶)) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) |
6 | 1, 5 | mpbid 146 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶)) → ¬ 𝐵 < 𝐴) |
7 | 6 | pm2.21d 609 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶)) → (𝐵 < 𝐴 → 𝐴 < 𝐶)) |
8 | idd 21 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶)) → (𝐴 < 𝐶 → 𝐴 < 𝐶)) | |
9 | simprr 522 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 < 𝐶) | |
10 | simpl3 991 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶)) → 𝐶 ∈ ℝ*) | |
11 | xrltso 9723 | . . . . . 6 ⊢ < Or ℝ* | |
12 | sowlin 4292 | . . . . . 6 ⊢ (( < Or ℝ* ∧ (𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝐴 ∈ ℝ*)) → (𝐵 < 𝐶 → (𝐵 < 𝐴 ∨ 𝐴 < 𝐶))) | |
13 | 11, 12 | mpan 421 | . . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐵 < 𝐶 → (𝐵 < 𝐴 ∨ 𝐴 < 𝐶))) |
14 | 3, 10, 2, 13 | syl3anc 1227 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶)) → (𝐵 < 𝐶 → (𝐵 < 𝐴 ∨ 𝐴 < 𝐶))) |
15 | 9, 14 | mpd 13 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶)) → (𝐵 < 𝐴 ∨ 𝐴 < 𝐶)) |
16 | 7, 8, 15 | mpjaod 708 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶)) → 𝐴 < 𝐶) |
17 | 16 | ex 114 | 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 967 ∈ wcel 2135 class class class wbr 3976 Or wor 4267 ℝ*cxr 7923 < clt 7924 ≤ cle 7925 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-po 4268 df-iso 4269 df-xp 4604 df-cnv 4606 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 |
This theorem is referenced by: xrlelttrd 9737 xrre 9747 xrre2 9748 iooss1 9843 iccssioo 9869 iccssico 9872 iocssioo 9890 ioossioo 9892 ico0 10187 bldisj 12942 xblm 12958 blsscls2 13034 metcnpi3 13058 |
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