Theorem poirr 5046

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 1039	. . 3 |- ( ( B e. A /\ B e. A /\ B e. A ) <-> ( ( B e. A /\ B e. A ) /\ B e. A ) )
2		anabs1 850	. . 3 |- ( ( ( B e. A /\ B e. A ) /\ B e. A ) <-> ( B e. A /\ B e. A ) )
3		anidm 676	. . 3 |- ( ( B e. A /\ B e. A ) <-> B e. A )
4	1, 2, 3	3bitrri 287	. 2 |- ( B e. A <-> ( B e. A /\ B e. A /\ B e. A ) )
5		pocl 5042	. . . 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 445	. . 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 475	. 2 |- ( ( R Po A /\ ( B e. A /\ B e. A /\ B e. A ) ) -> -. B R B )
8	4, 7	sylan2b 492	1 |- ( ( R Po A /\ B e. A ) -> -. B R B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   e. wcel 1990   class class class wbr 4653   Po wpo 5033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-po 5035

This theorem is referenced by:  po2nr  5048  pofun  5051  sonr  5056  poirr2  5520  predpoirr  5708  soisoi  6578  poxp  7289  swoer  7772  frfi  8205  wemappo  8454  zorn2lem3  9320  ex-po  27292  pocnv  31653  poseq  31750  ipo0  38653