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Theorem poirr 4777
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 984 . . 3  |-  ( ( B  e.  A  /\  B  e.  A  /\  B  e.  A )  <->  ( ( B  e.  A  /\  B  e.  A
)  /\  B  e.  A ) )
2 anabs1 815 . . 3  |-  ( ( ( B  e.  A  /\  B  e.  A
)  /\  B  e.  A )  <->  ( B  e.  A  /\  B  e.  A ) )
3 anidm 648 . . 3  |-  ( ( B  e.  A  /\  B  e.  A )  <->  B  e.  A )
41, 2, 33bitrri 275 . 2  |-  ( B  e.  A  <->  ( B  e.  A  /\  B  e.  A  /\  B  e.  A ) )
5 pocl 4773 . . . 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 430 . . 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 460 . 2  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  B  e.  A  /\  B  e.  A
) )  ->  -.  B R B )
84, 7sylan2b 477 1  |-  ( ( R  Po  A  /\  B  e.  A )  ->  -.  B R B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982    e. wcel 1867   class class class wbr 4417    Po wpo 4764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ral 2778  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-po 4766
This theorem is referenced by:  po2nr  4779  pofun  4782  sonr  4787  poirr2  5235  predpoirr  5418  soisoi  6225  poxp  6910  swoer  7390  frfi  7813  wemappo  8055  zorn2lem3  8917  ex-po  25756  pocnv  30217  poseq  30304  ipo0  36472
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