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Theorem sonr 4527
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 4523 . 2  |-  ( R  Or  A  ->  R  Po  A )
2 poirr 4517 . 2  |-  ( ( R  Po  A  /\  B  e.  A )  ->  -.  B R B )
31, 2sylan 459 1  |-  ( ( R  Or  A  /\  B  e.  A )  ->  -.  B R B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    e. wcel 1726   class class class wbr 4215    Po wpo 4504    Or wor 4505
This theorem is referenced by:  sotric  4532  sotrieq  4533  soirri  5263  soirriOLD  5268  suppr  7476  hartogslem1  7514  canth4  8527  canthwelem  8530  pwfseqlem4  8542  1ne0sr  8976  ltnr  9173  opsrtoslem2  16550  sltirr  25630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-po 4506  df-so 4507
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