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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 4719 | . 2 ⊢ ¬ 𝐴𝑅𝐴 |
4 | 1, 2 | sotri 4720 | . 2 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴𝑅𝐴) |
5 | 3, 4 | mto 588 | 1 ⊢ ¬ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 97 ⊆ wss 2917 class class class wbr 3764 Or wor 4032 × cxp 4343 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-po 4033 df-iso 4034 df-xp 4351 |
This theorem is referenced by: nqprdisj 6642 ltexprlemdisj 6704 recexprlemdisj 6728 caucvgprlemnkj 6764 caucvgprprlemnkltj 6787 caucvgprprlemnkeqj 6788 caucvgprprlemnjltk 6789 |
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