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Theorem poirr 4769
Description: A partial order relation is irreflexive. (Contributed by NM, 27-Mar-1997.)
Assertion
Ref Expression
poirr  |-  ( ( R  Po  A  /\  B  e.  A )  ->  -.  B R B )

Proof of Theorem poirr
StepHypRef Expression
1 df-3an 988 . . 3  |-  ( ( B  e.  A  /\  B  e.  A  /\  B  e.  A )  <->  ( ( B  e.  A  /\  B  e.  A
)  /\  B  e.  A ) )
2 anabs1 818 . . 3  |-  ( ( ( B  e.  A  /\  B  e.  A
)  /\  B  e.  A )  <->  ( B  e.  A  /\  B  e.  A ) )
3 anidm 650 . . 3  |-  ( ( B  e.  A  /\  B  e.  A )  <->  B  e.  A )
41, 2, 33bitrri 276 . 2  |-  ( B  e.  A  <->  ( B  e.  A  /\  B  e.  A  /\  B  e.  A ) )
5 pocl 4765 . . . 4  |-  ( R  Po  A  ->  (
( B  e.  A  /\  B  e.  A  /\  B  e.  A
)  ->  ( -.  B R B  /\  (
( B R B  /\  B R B )  ->  B R B ) ) ) )
65imp 431 . . 3  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  B  e.  A  /\  B  e.  A
) )  ->  ( -.  B R B  /\  ( ( B R B  /\  B R B )  ->  B R B ) ) )
76simpld 461 . 2  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  B  e.  A  /\  B  e.  A
) )  ->  -.  B R B )
84, 7sylan2b 478 1  |-  ( ( R  Po  A  /\  B  e.  A )  ->  -.  B R B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    /\ w3a 986    e. wcel 1889   class class class wbr 4405    Po wpo 4756
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ral 2744  df-rab 2748  df-v 3049  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-br 4406  df-po 4758
This theorem is referenced by:  po2nr  4771  pofun  4774  sonr  4779  poirr2  5227  predpoirr  5411  soisoi  6224  poxp  6913  swoer  7396  frfi  7821  wemappo  8069  zorn2lem3  8933  ex-po  25897  pocnv  30416  poseq  30503  ipo0  36813
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