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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 4933 | . 2 ⊢ ¬ 𝐴𝑅𝐴 |
4 | 1, 2 | sotri 4934 | . 2 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴𝑅𝐴) |
5 | 3, 4 | mto 651 | 1 ⊢ ¬ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ⊆ wss 3071 class class class wbr 3929 Or wor 4217 × cxp 4537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-po 4218 df-iso 4219 df-xp 4545 |
This theorem is referenced by: nqprdisj 7352 ltexprlemdisj 7414 recexprlemdisj 7438 caucvgprlemnkj 7474 caucvgprprlemnkltj 7497 caucvgprprlemnkeqj 7498 caucvgprprlemnjltk 7499 |
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