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| Description: A strict order relation has no 2-cycle loops. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.) | 
| Ref | Expression | 
|---|---|
| soi.1 | ⊢ 𝑅 Or 𝑆 | 
| soi.2 | ⊢ 𝑅 ⊆ (𝑆 × 𝑆) | 
| Ref | Expression | 
|---|---|
| son2lpi | ⊢ ¬ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | soi.1 | . . 3 ⊢ 𝑅 Or 𝑆 | |
| 2 | soi.2 | . . 3 ⊢ 𝑅 ⊆ (𝑆 × 𝑆) | |
| 3 | 1, 2 | soirri 6145 | . 2 ⊢ ¬ 𝐴𝑅𝐴 | 
| 4 | 1, 2 | sotri 6146 | . 2 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴𝑅𝐴) | 
| 5 | 3, 4 | mto 197 | 1 ⊢ ¬ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ∧ wa 395 ⊆ wss 3950 class class class wbr 5142 Or wor 5590 × cxp 5682 | 
| 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 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-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 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-opab 5205 df-po 5591 df-so 5592 df-xp 5690 | 
| This theorem is referenced by: (None) | 
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