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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 4345 | . 2 ⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴) | |
2 | po2nr 4341 | . 2 ⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵)) | |
3 | 1, 2 | sylan 283 | 1 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∈ wcel 2164 class class class wbr 4030 Po wpo 4326 Or wor 4327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2175 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-v 2762 df-un 3158 df-sn 3625 df-pr 3626 df-op 3628 df-br 4031 df-po 4328 df-iso 4329 |
This theorem is referenced by: sotricim 4355 cauappcvgprlemdisj 7713 cauappcvgprlemladdru 7718 cauappcvgprlemladdrl 7719 caucvgprlemnbj 7729 caucvgprprlemnbj 7755 suplocexprlemmu 7780 ltnsym2 8112 |
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