Theorem poirr 4447

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 1011	. . 3 |- ( ( B e. A /\ B e. A /\ B e. A ) <-> ( ( B e. A /\ B e. A ) /\ B e. A ) )
2		anabs1 578	. . 3 |- ( ( ( B e. A /\ B e. A ) /\ B e. A ) <-> ( B e. A /\ B e. A ) )
3		anidm 400	. . 3 |- ( ( B e. A /\ B e. A ) <-> B e. A )
4	1, 2, 3	3bitrri 207	. 2 |- ( B e. A <-> ( B e. A /\ B e. A /\ B e. A ) )
5		pocl 4443	. . . 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 124	. . 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 112	. 2 |- ( ( R Po A /\ ( B e. A /\ B e. A /\ B e. A ) ) -> -. B R B )
8	4, 7	sylan2b 287	1 |- ( ( R Po A /\ B e. A ) -> -. B R B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 1009   e. wcel 2209   class class class wbr 4125   Po wpo 4434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-po 4436

This theorem is referenced by:  po2nr  4449  pofun  4452  sonr  4457  poirr2  5175  poxp  6458  swoer  6825  tridc  7194  fimax2gtrilemstep  7195