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Mirrors > Home > ILE Home > Th. List > xrlelttrd | GIF version |
Description: Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.) |
Ref | Expression |
---|---|
xrlttrd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
xrlttrd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
xrlttrd.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ*) |
xrlelttrd.4 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
xrlelttrd.5 | ⊢ (𝜑 → 𝐵 < 𝐶) |
Ref | Expression |
---|---|
xrlelttrd | ⊢ (𝜑 → 𝐴 < 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrlelttrd.4 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
2 | xrlelttrd.5 | . 2 ⊢ (𝜑 → 𝐵 < 𝐶) | |
3 | xrlttrd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
4 | xrlttrd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
5 | xrlttrd.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ*) | |
6 | xrlelttr 9831 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) | |
7 | 3, 4, 5, 6 | syl3anc 1249 | . 2 ⊢ (𝜑 → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) |
8 | 1, 2, 7 | mp2and 433 | 1 ⊢ (𝜑 → 𝐴 < 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2160 class class class wbr 4018 ℝ*cxr 8016 < clt 8017 ≤ cle 8018 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7927 ax-resscn 7928 ax-pre-ltirr 7948 ax-pre-ltwlin 7949 ax-pre-lttrn 7950 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-po 4311 df-iso 4312 df-xp 4647 df-cnv 4649 df-pnf 8019 df-mnf 8020 df-xr 8021 df-ltxr 8022 df-le 8023 |
This theorem is referenced by: xlt2add 9905 elioc2 9961 elicc2 9963 xrmaxltsup 11293 blgt0 14339 xblss2ps 14341 xblss2 14342 tgioo 14483 |
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