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Theorem poirr 4125
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 926 . . 3 ((𝐵𝐴𝐵𝐴𝐵𝐴) ↔ ((𝐵𝐴𝐵𝐴) ∧ 𝐵𝐴))
2 anabs1 539 . . 3 (((𝐵𝐴𝐵𝐴) ∧ 𝐵𝐴) ↔ (𝐵𝐴𝐵𝐴))
3 anidm 388 . . 3 ((𝐵𝐴𝐵𝐴) ↔ 𝐵𝐴)
41, 2, 33bitrri 205 . 2 (𝐵𝐴 ↔ (𝐵𝐴𝐵𝐴𝐵𝐴))
5 pocl 4121 . . . 4 (𝑅 Po 𝐴 → ((𝐵𝐴𝐵𝐴𝐵𝐴) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐵𝐵𝑅𝐵) → 𝐵𝑅𝐵))))
65imp 122 . . 3 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐵𝐴𝐵𝐴)) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐵𝐵𝑅𝐵) → 𝐵𝑅𝐵)))
76simpld 110 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐵𝐴𝐵𝐴)) → ¬ 𝐵𝑅𝐵)
84, 7sylan2b 281 1 ((𝑅 Po 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  w3a 924  wcel 1438   class class class wbr 3837   Po wpo 4112
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-po 4114
This theorem is referenced by:  po2nr  4127  pofun  4130  sonr  4135  poirr2  4811  poxp  5979  swoer  6300  tridc  6595  fimax2gtrilemstep  6596
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