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Theorem poirr 4514
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 938 . . 3  |-  ( ( B  e.  A  /\  B  e.  A  /\  B  e.  A )  <->  ( ( B  e.  A  /\  B  e.  A
)  /\  B  e.  A ) )
2 anabs1 784 . . 3  |-  ( ( ( B  e.  A  /\  B  e.  A
)  /\  B  e.  A )  <->  ( B  e.  A  /\  B  e.  A ) )
3 anidm 626 . . 3  |-  ( ( B  e.  A  /\  B  e.  A )  <->  B  e.  A )
41, 2, 33bitrri 264 . 2  |-  ( B  e.  A  <->  ( B  e.  A  /\  B  e.  A  /\  B  e.  A ) )
5 pocl 4510 . . . 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 419 . . 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 446 . 2  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  B  e.  A  /\  B  e.  A
) )  ->  -.  B R B )
84, 7sylan2b 462 1  |-  ( ( R  Po  A  /\  B  e.  A )  ->  -.  B R B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1725   class class class wbr 4212    Po wpo 4501
This theorem is referenced by:  po2nr  4516  pofun  4519  sonr  4524  poirr2  5258  soisoi  6048  poxp  6458  swoer  6933  frfi  7352  wemappo  7518  zorn2lem3  8378  ex-po  21743  pocnv  25387  predpoirr  25472  poseq  25528  ipo0  27628
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-po 4503
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