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| Description: A partial order has no 2-cycle loops. (Contributed by NM, 27-Mar-1997.) | 
| Ref | Expression | 
|---|---|
| po2nr | ⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | poirr 5603 | . . 3 ⊢ ((𝑅 Po 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵) | |
| 2 | 1 | adantrr 717 | . 2 ⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ 𝐵𝑅𝐵) | 
| 3 | potr 5604 | . . . . . 6 ⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴)) → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐵𝑅𝐵)) | |
| 4 | 3 | 3exp2 1354 | . . . . 5 ⊢ (𝑅 Po 𝐴 → (𝐵 ∈ 𝐴 → (𝐶 ∈ 𝐴 → (𝐵 ∈ 𝐴 → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐵𝑅𝐵))))) | 
| 5 | 4 | com34 91 | . . . 4 ⊢ (𝑅 Po 𝐴 → (𝐵 ∈ 𝐴 → (𝐵 ∈ 𝐴 → (𝐶 ∈ 𝐴 → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐵𝑅𝐵))))) | 
| 6 | 5 | pm2.43d 53 | . . 3 ⊢ (𝑅 Po 𝐴 → (𝐵 ∈ 𝐴 → (𝐶 ∈ 𝐴 → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐵𝑅𝐵)))) | 
| 7 | 6 | imp32 418 | . 2 ⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐵𝑅𝐵)) | 
| 8 | 2, 7 | mtod 198 | 1 ⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∈ wcel 2107 class class class wbr 5142 Po wpo 5589 | 
| 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-ext 2707 | 
| 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-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-po 5591 | 
| This theorem is referenced by: po3nr 5606 so2nr 5619 soisoi 7349 poxp2 8169 poxp3 8176 wemaplem2 9588 pospo 18391 poprelb 47516 | 
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