MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sonr Structured version   Unicode version

Theorem sonr 4827
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 4823 . 2  |-  ( R  Or  A  ->  R  Po  A )
2 poirr 4817 . 2  |-  ( ( R  Po  A  /\  B  e.  A )  ->  -.  B R B )
31, 2sylan 471 1  |-  ( ( R  Or  A  /\  B  e.  A )  ->  -.  B R B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    e. wcel 1767   class class class wbr 4453    Po wpo 4804    Or wor 4805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-po 4806  df-so 4807
This theorem is referenced by:  sotric  4832  sotrieq  4833  soirri  5399  soirriOLD  5404  suppr  7941  hartogslem1  7979  canth4  9037  canthwelem  9040  pwfseqlem4  9052  1ne0sr  9485  ltnr  9691  opsrtoslem2  18019  sltirr  29357  fin2solem  29966  fin2so  29967
  Copyright terms: Public domain W3C validator