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Theorem sonr 4980
 Description: A strict order relation is irreflexive. (Contributed by NM, 24-Nov-1995.)
Assertion
Ref Expression
sonr ((𝑅 Or 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)

Proof of Theorem sonr
StepHypRef Expression
1 sopo 4976 . 2 (𝑅 Or 𝐴𝑅 Po 𝐴)
2 poirr 4970 . 2 ((𝑅 Po 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
31, 2sylan 487 1 ((𝑅 Or 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∈ wcel 1977   class class class wbr 4583   Po wpo 4957   Or wor 4958 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-po 4959  df-so 4960 This theorem is referenced by:  sotric  4985  sotrieq  4986  soirri  5441  suppr  8260  infpr  8292  hartogslem1  8330  canth4  9348  canthwelem  9351  pwfseqlem4  9363  1ne0sr  9796  ltnr  10011  opsrtoslem2  19306  sltirr  31069  fin2solem  32565  fin2so  32566
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