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Theorem po2nr 4037
 Description: A partial order relation has no 2-cycle loops. (Contributed by NM, 27-Mar-1997.)
Assertion
Ref Expression
po2nr ((𝑅 Po A (B A 𝐶 A)) → ¬ (B𝑅𝐶 𝐶𝑅B))

Proof of Theorem po2nr
StepHypRef Expression
1 poirr 4035 . . 3 ((𝑅 Po A B A) → ¬ B𝑅B)
21adantrr 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))
433exp2 1121 . . . . 5 (𝑅 Po A → (B A → (𝐶 A → (B A → ((B𝑅𝐶 𝐶𝑅B) → B𝑅B)))))
54com34 77 . . . 4 (𝑅 Po A → (B A → (B A → (𝐶 A → ((B𝑅𝐶 𝐶𝑅B) → B𝑅B)))))
65pm2.43d 44 . . 3 (𝑅 Po A → (B A → (𝐶 A → ((B𝑅𝐶 𝐶𝑅B) → B𝑅B))))
76imp32 244 . 2 ((𝑅 Po A (B A 𝐶 A)) → ((B𝑅𝐶 𝐶𝑅B) → B𝑅B))
82, 7mtod 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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