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Theorem sonr 4488
Description: A strict order relation is irreflexive. (Contributed by NM, 24-Nov-1995.)
Assertion
Ref Expression
sonr  |-  ( ( R  Or  A  /\  B  e.  A )  ->  -.  B R B )

Proof of Theorem sonr
StepHypRef Expression
1 sopo 4484 . 2  |-  ( R  Or  A  ->  R  Po  A )
2 poirr 4478 . 2  |-  ( ( R  Po  A  /\  B  e.  A )  ->  -.  B R B )
31, 2sylan 458 1  |-  ( ( R  Or  A  /\  B  e.  A )  ->  -.  B R B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    e. wcel 1721   class class class wbr 4176    Po wpo 4465    Or wor 4466
This theorem is referenced by:  sotric  4493  sotrieq  4494  soirri  5223  soirriOLD  5228  suppr  7433  hartogslem1  7471  canth4  8482  canthwelem  8485  pwfseqlem4  8497  1ne0sr  8931  ltnr  9128  opsrtoslem2  16504  sltirr  25542
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ral 2675  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-po 4467  df-so 4468
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