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Theorem po2nr 4309
Description: A partial order relation has no 2-cycle loops. (Contributed by NM, 27-Mar-1997.)
Assertion
Ref Expression
po2nr ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))

Proof of Theorem po2nr
StepHypRef Expression
1 poirr 4307 . . 3 ((𝑅 Po 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
21adantrr 479 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ 𝐵𝑅𝐵)
3 potr 4308 . . . . . 6 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐵))
433exp2 1225 . . . . 5 (𝑅 Po 𝐴 → (𝐵𝐴 → (𝐶𝐴 → (𝐵𝐴 → ((𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐵)))))
54com34 83 . . . 4 (𝑅 Po 𝐴 → (𝐵𝐴 → (𝐵𝐴 → (𝐶𝐴 → ((𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐵)))))
65pm2.43d 50 . . 3 (𝑅 Po 𝐴 → (𝐵𝐴 → (𝐶𝐴 → ((𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐵))))
76imp32 257 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐵) → 𝐵𝑅𝐵))
82, 7mtod 663 1 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wcel 2148   class class class wbr 4003   Po wpo 4294
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-po 4296
This theorem is referenced by:  po3nr  4310  so2nr  4321  tridc  6898
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