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Theorem sonr 5612
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 5608 . 2 (𝑅 Or 𝐴𝑅 Po 𝐴)
2 poirr 5601 . 2 ((𝑅 Po 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
31, 2sylan 581 1 ((𝑅 Or 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wcel 2107   class class class wbr 5149   Po wpo 5587   Or wor 5588
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 2704
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 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-po 5589  df-so 5590
This theorem is referenced by:  sotric  5617  sotrieq  5618  soirri  6128  suppr  9466  infpr  9498  hartogslem1  9537  canth4  10642  canthwelem  10645  pwfseqlem4  10657  1ne0sr  11091  ltnr  11309  opsrtoslem2  21617  nodenselem4  27190  nodenselem5  27191  nodenselem7  27193  nolt02o  27198  nogt01o  27199  noresle  27200  nosupbnd1lem1  27211  nosupbnd2lem1  27218  noinfbnd1lem1  27226  noinfbnd2lem1  27233  sltirr  27249  fin2solem  36474  fin2so  36475
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