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Mirrors > Home > ILE Home > Th. List > po2nr | GIF version |
Description: A partial order relation has no 2-cycle loops. (Contributed by NM, 27-Mar-1997.) |
Ref | Expression |
---|---|
po2nr | ⊢ ((𝑅 Po A ∧ (B ∈ A ∧ 𝐶 ∈ A)) → ¬ (B𝑅𝐶 ∧ 𝐶𝑅B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | poirr 4035 | . . 3 ⊢ ((𝑅 Po A ∧ B ∈ A) → ¬ B𝑅B) | |
2 | 1 | adantrr 448 | . 2 ⊢ ((𝑅 Po A ∧ (B ∈ A ∧ 𝐶 ∈ A)) → ¬ B𝑅B) |
3 | potr 4036 | . . . . . 6 ⊢ ((𝑅 Po A ∧ (B ∈ A ∧ 𝐶 ∈ A ∧ B ∈ A)) → ((B𝑅𝐶 ∧ 𝐶𝑅B) → B𝑅B)) | |
4 | 3 | 3exp2 1121 | . . . . 5 ⊢ (𝑅 Po A → (B ∈ A → (𝐶 ∈ A → (B ∈ A → ((B𝑅𝐶 ∧ 𝐶𝑅B) → B𝑅B))))) |
5 | 4 | com34 77 | . . . 4 ⊢ (𝑅 Po A → (B ∈ A → (B ∈ A → (𝐶 ∈ A → ((B𝑅𝐶 ∧ 𝐶𝑅B) → B𝑅B))))) |
6 | 5 | pm2.43d 44 | . . 3 ⊢ (𝑅 Po A → (B ∈ A → (𝐶 ∈ A → ((B𝑅𝐶 ∧ 𝐶𝑅B) → B𝑅B)))) |
7 | 6 | imp32 244 | . 2 ⊢ ((𝑅 Po A ∧ (B ∈ A ∧ 𝐶 ∈ A)) → ((B𝑅𝐶 ∧ 𝐶𝑅B) → B𝑅B)) |
8 | 2, 7 | mtod 588 | 1 ⊢ ((𝑅 Po A ∧ (B ∈ A ∧ 𝐶 ∈ A)) → ¬ (B𝑅𝐶 ∧ 𝐶𝑅B)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ∈ wcel 1390 class class class wbr 3755 Po wpo 4022 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-v 2553 df-un 2916 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-po 4024 |
This theorem is referenced by: po3nr 4038 so2nr 4049 |
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