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| Description: A poset ordering is transitive. (Contributed by NM, 11-Sep-2011.) | 
| Ref | Expression | 
|---|---|
| posi.b | ⊢ 𝐵 = (Base‘𝐾) | 
| posi.l | ⊢ ≤ = (le‘𝐾) | 
| Ref | Expression | 
|---|---|
| postr | ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | posi.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
| 2 | posi.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
| 3 | 1, 2 | posi 18363 | . 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) → 𝑋 = 𝑌) ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍))) | 
| 4 | 3 | simp3d 1145 | 1 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 Basecbs 17247 lecple 17304 Posetcpo 18353 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-poset 18359 | 
| This theorem is referenced by: odupos 18373 plttr 18387 joinle 18431 meetle 18445 lattr 18489 omndadd2d 33085 omndadd2rd 33086 omndmul2 33089 atlatle 39321 cvratlem 39423 llncmp 39524 llncvrlpln 39560 lplncmp 39564 lplncvrlvol 39618 lvolcmp 39619 pmaple 39763 | 
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