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Mirrors > Home > ILE Home > Th. List > son2lpi | 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 4662 | . 2 ⊢ ¬ A𝑅A |
4 | 1, 2 | sotri 4663 | . 2 ⊢ ((A𝑅B ∧ B𝑅A) → A𝑅A) |
5 | 3, 4 | mto 587 | 1 ⊢ ¬ (A𝑅B ∧ B𝑅A) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 97 ⊆ wss 2911 class class class wbr 3755 Or wor 4023 × cxp 4286 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-po 4024 df-iso 4025 df-xp 4294 |
This theorem is referenced by: nqprdisj 6527 ltexprlemdisj 6580 recexprlemdisj 6602 caucvgprlemnkj 6637 |
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