Theorem poirr 4237

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

Step	Hyp	Ref	Expression
1		df-3an 965	. . 3 |- ( ( B e. A /\ B e. A /\ B e. A ) <-> ( ( B e. A /\ B e. A ) /\ B e. A ) )
2		anabs1 562	. . 3 |- ( ( ( B e. A /\ B e. A ) /\ B e. A ) <-> ( B e. A /\ B e. A ) )
3		anidm 394	. . 3 |- ( ( B e. A /\ B e. A ) <-> B e. A )
4	1, 2, 3	3bitrri 206	. 2 |- ( B e. A <-> ( B e. A /\ B e. A /\ B e. A ) )
5		pocl 4233	. . . 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 ) ) ) )
6	5	imp 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 ) ) )
7	6	simpld 111	. 2 |- ( ( R Po A /\ ( B e. A /\ B e. A /\ B e. A ) ) -> -. B R B )
8	4, 7	sylan2b 285	1 |- ( ( R Po A /\ B e. A ) -> -. B R B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   /\ w3a 963   e. wcel 1481   class class class wbr 3937   Po wpo 4224

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-po 4226

This theorem is referenced by:  po2nr  4239  pofun  4242  sonr  4247  poirr2  4939  poxp  6137  swoer  6465  tridc  6801  fimax2gtrilemstep  6802