Theorem po3nr 4328

Description: A partial order relation has no 3-cycle loops. (Contributed by NM, 27-Mar-1997.)

Assertion

Ref	Expression
po3nr	|- ( ( R Po A /\ ( B e. A /\ C e. A /\ D e. A ) ) -> -. ( B R C /\ C R D /\ D R B ) )

Proof of Theorem po3nr

Step	Hyp	Ref	Expression
1		po2nr 4327	. . 3 |- ( ( R Po A /\ ( B e. A /\ D e. A ) ) -> -. ( B R D /\ D R B ) )
2	1	3adantr2 1159	. 2 |- ( ( R Po A /\ ( B e. A /\ C e. A /\ D e. A ) ) -> -. ( B R D /\ D R B ) )
3		df-3an 982	. . 3 |- ( ( B R C /\ C R D /\ D R B ) <-> ( ( B R C /\ C R D ) /\ D R B ) )
4		potr 4326	. . . 4 |- ( ( R Po A /\ ( B e. A /\ C e. A /\ D e. A ) ) -> ( ( B R C /\ C R D ) -> B R D ) )
5	4	anim1d 336	. . 3 |- ( ( R Po A /\ ( B e. A /\ C e. A /\ D e. A ) ) -> ( ( ( B R C /\ C R D ) /\ D R B ) -> ( B R D /\ D R B ) ) )
6	3, 5	biimtrid 152	. 2 |- ( ( R Po A /\ ( B e. A /\ C e. A /\ D e. A ) ) -> ( ( B R C /\ C R D /\ D R B ) -> ( B R D /\ D R B ) ) )
7	2, 6	mtod 664	1 |- ( ( R Po A /\ ( B e. A /\ C e. A /\ D e. A ) ) -> -. ( B R C /\ C R D /\ D R B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 980   e. wcel 2160   class class class wbr 4018   Po wpo 4312

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-po 4314

This theorem is referenced by:  so3nr  4340