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Theorem poirr 4064
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 922 . . 3  |-  ( ( B  e.  A  /\  B  e.  A  /\  B  e.  A )  <->  ( ( B  e.  A  /\  B  e.  A
)  /\  B  e.  A ) )
2 anabs1 537 . . 3  |-  ( ( ( B  e.  A  /\  B  e.  A
)  /\  B  e.  A )  <->  ( B  e.  A  /\  B  e.  A ) )
3 anidm 388 . . 3  |-  ( ( B  e.  A  /\  B  e.  A )  <->  B  e.  A )
41, 2, 33bitrri 205 . 2  |-  ( B  e.  A  <->  ( B  e.  A  /\  B  e.  A  /\  B  e.  A ) )
5 pocl 4060 . . . 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 122 . . 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 110 . 2  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  B  e.  A  /\  B  e.  A
) )  ->  -.  B R B )
84, 7sylan2b 281 1  |-  ( ( R  Po  A  /\  B  e.  A )  ->  -.  B R B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    /\ w3a 920    e. wcel 1434   class class class wbr 3787    Po wpo 4051
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604  df-un 2978  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-po 4053
This theorem is referenced by:  po2nr  4066  pofun  4069  sonr  4074  poirr2  4741  poxp  5878  swoer  6193
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