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Mirrors > Home > MPE Home > Th. List > xrlttrd | Structured version Visualization version 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*) |
xrlttrd.4 | ⊢ (𝜑 → 𝐴 < 𝐵) |
xrlttrd.5 | ⊢ (𝜑 → 𝐵 < 𝐶) |
Ref | Expression |
---|---|
xrlttrd | ⊢ (𝜑 → 𝐴 < 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrlttrd.4 | . 2 ⊢ (𝜑 → 𝐴 < 𝐵) | |
2 | xrlttrd.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 | xrlttr 12521 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) | |
7 | 3, 4, 5, 6 | syl3anc 1368 | . 2 ⊢ (𝜑 → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) |
8 | 1, 2, 7 | mp2and 698 | 1 ⊢ (𝜑 → 𝐴 < 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 class class class wbr 5030 ℝ*cxr 10663 < clt 10664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 |
This theorem is referenced by: xlt2add 12641 qdensere 23375 lmnn 23867 dvferm1lem 24587 itgsubst 24652 pserdvlem1 25022 pserdvlem2 25023 iooelexlt 34779 heicant 35092 itg2gt0cn 35112 imo72b2 40878 suplesup 41971 infxr 41999 infleinflem2 42003 ioorrnopnxrlem 42948 omeiunltfirp 43158 bgoldbtbnd 44327 |
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