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Mirrors > Home > ILE Home > Th. List > so2nr | GIF version |
Description: A strict order relation has no 2-cycle loops. (Contributed by NM, 21-Jan-1996.) |
Ref | Expression |
---|---|
so2nr | ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sopo 4050 | . 2 ⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴) | |
2 | po2nr 4046 | . 2 ⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵)) | |
3 | 1, 2 | sylan 267 | 1 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ∈ wcel 1393 class class class wbr 3764 Po wpo 4031 Or wor 4032 |
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-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
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-v 2559 df-un 2922 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-po 4033 df-iso 4034 |
This theorem is referenced by: sotricim 4060 cauappcvgprlemdisj 6749 cauappcvgprlemladdru 6754 cauappcvgprlemladdrl 6755 caucvgprlemnbj 6765 caucvgprprlemnbj 6791 ltnsym2 7108 |
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