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Mirrors > Home > MPE Home > Th. List > son2lpi | Structured version Visualization version GIF version |
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 5988 | . 2 ⊢ ¬ 𝐴𝑅𝐴 |
4 | 1, 2 | sotri 5989 | . 2 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴𝑅𝐴) |
5 | 3, 4 | mto 199 | 1 ⊢ ¬ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 ⊆ wss 3938 class class class wbr 5068 Or wor 5475 × cxp 5555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-po 5476 df-so 5477 df-xp 5563 |
This theorem is referenced by: (None) |
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