|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > pltn2lp | Structured version Visualization version GIF version | ||
| Description: The less-than relation has no 2-cycle loops. (pssn2lp 4103 analog.) (Contributed by NM, 2-Dec-2011.) | 
| Ref | Expression | 
|---|---|
| pltnlt.b | ⊢ 𝐵 = (Base‘𝐾) | 
| pltnlt.s | ⊢ < = (lt‘𝐾) | 
| Ref | Expression | 
|---|---|
| pltn2lp | ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ¬ (𝑋 < 𝑌 ∧ 𝑌 < 𝑋)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | pltnlt.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
| 2 | eqid 2736 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
| 3 | pltnlt.s | . . . . 5 ⊢ < = (lt‘𝐾) | |
| 4 | 1, 2, 3 | pltnle 18384 | . . . 4 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → ¬ 𝑌(le‘𝐾)𝑋) | 
| 5 | 4 | ex 412 | . . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 → ¬ 𝑌(le‘𝐾)𝑋)) | 
| 6 | 2, 3 | pltle 18379 | . . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑌 < 𝑋 → 𝑌(le‘𝐾)𝑋)) | 
| 7 | 6 | 3com23 1126 | . . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 < 𝑋 → 𝑌(le‘𝐾)𝑋)) | 
| 8 | 5, 7 | nsyld 156 | . 2 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 → ¬ 𝑌 < 𝑋)) | 
| 9 | imnan 399 | . 2 ⊢ ((𝑋 < 𝑌 → ¬ 𝑌 < 𝑋) ↔ ¬ (𝑋 < 𝑌 ∧ 𝑌 < 𝑋)) | |
| 10 | 8, 9 | sylib 218 | 1 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ¬ (𝑋 < 𝑌 ∧ 𝑌 < 𝑋)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 class class class wbr 5142 ‘cfv 6560 Basecbs 17248 lecple 17305 Posetcpo 18354 ltcplt 18355 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-proset 18341 df-poset 18360 df-plt 18376 | 
| This theorem is referenced by: plttr 18388 | 
| Copyright terms: Public domain | W3C validator |