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Theorem sonr 5610
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 5606 . 2 (𝑅 Or 𝐴𝑅 Po 𝐴)
2 poirr 5599 . 2 ((𝑅 Po 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
31, 2sylan 580 1 ((𝑅 Or 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wcel 2106   class class class wbr 5147   Po wpo 5585   Or wor 5586
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  df-so 5588
This theorem is referenced by:  sotric  5615  sotrieq  5616  soirri  6124  suppr  9462  infpr  9494  hartogslem1  9533  canth4  10638  canthwelem  10641  pwfseqlem4  10653  1ne0sr  11087  ltnr  11305  opsrtoslem2  21608  nodenselem4  27179  nodenselem5  27180  nodenselem7  27182  nolt02o  27187  nogt01o  27188  noresle  27189  nosupbnd1lem1  27200  nosupbnd2lem1  27207  noinfbnd1lem1  27215  noinfbnd2lem1  27222  sltirr  27238  fin2solem  36462  fin2so  36463
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