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Theorem poirr 5599
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 1089 . . 3 ((𝐵𝐴𝐵𝐴𝐵𝐴) ↔ ((𝐵𝐴𝐵𝐴) ∧ 𝐵𝐴))
2 anabs1 660 . . 3 (((𝐵𝐴𝐵𝐴) ∧ 𝐵𝐴) ↔ (𝐵𝐴𝐵𝐴))
3 anidm 565 . . 3 ((𝐵𝐴𝐵𝐴) ↔ 𝐵𝐴)
41, 2, 33bitrri 297 . 2 (𝐵𝐴 ↔ (𝐵𝐴𝐵𝐴𝐵𝐴))
5 pocl 5594 . . . 4 (𝑅 Po 𝐴 → ((𝐵𝐴𝐵𝐴𝐵𝐴) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐵𝐵𝑅𝐵) → 𝐵𝑅𝐵))))
65imp 407 . . 3 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐵𝐴𝐵𝐴)) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐵𝐵𝑅𝐵) → 𝐵𝑅𝐵)))
76simpld 495 . 2 ((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐵𝐴𝐵𝐴)) → ¬ 𝐵𝑅𝐵)
84, 7sylan2b 594 1 ((𝑅 Po 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1087  wcel 2106   class class class wbr 5147   Po wpo 5585
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-po 5587
This theorem is referenced by:  po2nr  5601  po2ne  5603  pofun  5605  sonr  5610  poirr2  6122  predpoirr  6331  soisoi  7321  poxp  8110  poseq  8140  swoer  8729  frfi  9284  wemappo  9540  zorn2lem3  10489  ex-po  29677  pocnv  34721  epirron  41988  ipo0  43193
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