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Theorem poirr 5558
Description: A partial order 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 1090 . . 3 ((𝐵𝐴𝐵𝐴𝐵𝐴) ↔ ((𝐵𝐴𝐵𝐴) ∧ 𝐵𝐴))
2 anabs1 661 . . 3 (((𝐵𝐴𝐵𝐴) ∧ 𝐵𝐴) ↔ (𝐵𝐴𝐵𝐴))
3 anidm 566 . . 3 ((𝐵𝐴𝐵𝐴) ↔ 𝐵𝐴)
41, 2, 33bitrri 298 . 2 (𝐵𝐴 ↔ (𝐵𝐴𝐵𝐴𝐵𝐴))
5 pocl 5553 . . . 4 (𝑅 Po 𝐴 → ((𝐵𝐴𝐵𝐴𝐵𝐴) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐵𝐵𝑅𝐵) → 𝐵𝑅𝐵))))
65imp 408 . . 3 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐵𝐴𝐵𝐴)) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐵𝐵𝑅𝐵) → 𝐵𝑅𝐵)))
76simpld 496 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐵𝐴𝐵𝐴)) → ¬ 𝐵𝑅𝐵)
84, 7sylan2b 595 1 ((𝑅 Po 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  w3a 1088  wcel 2107   class class class wbr 5106   Po wpo 5544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-po 5546
This theorem is referenced by:  po2nr  5560  po2ne  5562  pofun  5564  sonr  5569  poirr2  6079  predpoirr  6288  soisoi  7274  poxp  8061  poseq  8091  swoer  8679  frfi  9233  wemappo  9486  zorn2lem3  10435  ex-po  29382  pocnv  34339  epirron  41591  ipo0  42736
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