Theorem po3nr 4345

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 4344	. . 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 4343	. . . 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 2167   class class class wbr 4033   Po wpo 4329

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-po 4331

This theorem is referenced by:  so3nr  4357