ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  poirr Unicode version

Theorem poirr 4199
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 949 . . 3  |-  ( ( B  e.  A  /\  B  e.  A  /\  B  e.  A )  <->  ( ( B  e.  A  /\  B  e.  A
)  /\  B  e.  A ) )
2 anabs1 546 . . 3  |-  ( ( ( B  e.  A  /\  B  e.  A
)  /\  B  e.  A )  <->  ( B  e.  A  /\  B  e.  A ) )
3 anidm 393 . . 3  |-  ( ( B  e.  A  /\  B  e.  A )  <->  B  e.  A )
41, 2, 33bitrri 206 . 2  |-  ( B  e.  A  <->  ( B  e.  A  /\  B  e.  A  /\  B  e.  A ) )
5 pocl 4195 . . . 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 123 . . 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 111 . 2  |-  ( ( R  Po  A  /\  ( B  e.  A  /\  B  e.  A  /\  B  e.  A
) )  ->  -.  B R B )
84, 7sylan2b 285 1  |-  ( ( R  Po  A  /\  B  e.  A )  ->  -.  B R B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    /\ w3a 947    e. wcel 1465   class class class wbr 3899    Po wpo 4186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-po 4188
This theorem is referenced by:  po2nr  4201  pofun  4204  sonr  4209  poirr2  4901  poxp  6097  swoer  6425  tridc  6761  fimax2gtrilemstep  6762
  Copyright terms: Public domain W3C validator