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Theorem poirr 5449
 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 1086 . . 3 ((𝐵𝐴𝐵𝐴𝐵𝐴) ↔ ((𝐵𝐴𝐵𝐴) ∧ 𝐵𝐴))
2 anabs1 661 . . 3 (((𝐵𝐴𝐵𝐴) ∧ 𝐵𝐴) ↔ (𝐵𝐴𝐵𝐴))
3 anidm 568 . . 3 ((𝐵𝐴𝐵𝐴) ↔ 𝐵𝐴)
41, 2, 33bitrri 301 . 2 (𝐵𝐴 ↔ (𝐵𝐴𝐵𝐴𝐵𝐴))
5 pocl 5445 . . . 4 (𝑅 Po 𝐴 → ((𝐵𝐴𝐵𝐴𝐵𝐴) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐵𝐵𝑅𝐵) → 𝐵𝑅𝐵))))
65imp 410 . . 3 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐵𝐴𝐵𝐴)) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐵𝐵𝑅𝐵) → 𝐵𝑅𝐵)))
76simpld 498 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐵𝐴𝐵𝐴)) → ¬ 𝐵𝑅𝐵)
84, 7sylan2b 596 1 ((𝑅 Po 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   ∈ wcel 2111   class class class wbr 5030   Po wpo 5436 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-v 3443  df-un 3886  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-po 5438 This theorem is referenced by:  po2nr  5451  po2ne  5453  pofun  5455  sonr  5460  poirr2  5951  predpoirr  6144  soisoi  7060  poxp  7805  swoer  8302  frfi  8747  wemappo  8997  zorn2lem3  9909  ex-po  28220  pocnv  33112  poseq  33208  ipo0  41151
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