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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 9332 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) | |
7 | 3, 4, 5, 6 | syl3anc 1175 | . 2 ⊢ (𝜑 → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) |
8 | 1, 2, 7 | mp2and 425 | 1 ⊢ (𝜑 → 𝐴 < 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1439 class class class wbr 3851 ℝ*cxr 7582 < clt 7583 ≤ cle 7584 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-cnex 7497 ax-resscn 7498 ax-pre-ltirr 7518 ax-pre-ltwlin 7519 ax-pre-lttrn 7520 |
This theorem depends on definitions: df-bi 116 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-rab 2369 df-v 2622 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-po 4132 df-iso 4133 df-xp 4458 df-cnv 4460 df-pnf 7585 df-mnf 7586 df-xr 7587 df-ltxr 7588 df-le 7589 |
This theorem is referenced by: elioc2 9415 elicc2 9417 |
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