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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 4781 | . 2 ⊢ ¬ 𝐴𝑅𝐴 |
4 | 1, 2 | sotri 4782 | . 2 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴𝑅𝐴) |
5 | 3, 4 | mto 621 | 1 ⊢ ¬ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 102 ⊆ wss 2984 class class class wbr 3811 Or wor 4086 × cxp 4399 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 4000 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-v 2614 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-br 3812 df-opab 3866 df-po 4087 df-iso 4088 df-xp 4407 |
This theorem is referenced by: nqprdisj 7006 ltexprlemdisj 7068 recexprlemdisj 7092 caucvgprlemnkj 7128 caucvgprprlemnkltj 7151 caucvgprprlemnkeqj 7152 caucvgprprlemnjltk 7153 |
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