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Theorem poirr 4325
Description: A partial order relation is irreflexive. (Contributed by NM, 27-Mar-1997.)
Assertion
Ref Expression
poirr ((𝑅 Po 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)

Proof of Theorem poirr
StepHypRef Expression
1 df-3an 982 . . 3 ((𝐵𝐴𝐵𝐴𝐵𝐴) ↔ ((𝐵𝐴𝐵𝐴) ∧ 𝐵𝐴))
2 anabs1 572 . . 3 (((𝐵𝐴𝐵𝐴) ∧ 𝐵𝐴) ↔ (𝐵𝐴𝐵𝐴))
3 anidm 396 . . 3 ((𝐵𝐴𝐵𝐴) ↔ 𝐵𝐴)
41, 2, 33bitrri 207 . 2 (𝐵𝐴 ↔ (𝐵𝐴𝐵𝐴𝐵𝐴))
5 pocl 4321 . . . 4 (𝑅 Po 𝐴 → ((𝐵𝐴𝐵𝐴𝐵𝐴) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐵𝐵𝑅𝐵) → 𝐵𝑅𝐵))))
65imp 124 . . 3 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐵𝐴𝐵𝐴)) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐵𝐵𝑅𝐵) → 𝐵𝑅𝐵)))
76simpld 112 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐵𝐴𝐵𝐴)) → ¬ 𝐵𝑅𝐵)
84, 7sylan2b 287 1 ((𝑅 Po 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  w3a 980  wcel 2160   class class class wbr 4018   Po wpo 4312
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 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-po 4314
This theorem is referenced by:  po2nr  4327  pofun  4330  sonr  4335  poirr2  5039  poxp  6256  swoer  6586  tridc  6926  fimax2gtrilemstep  6927
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