Theorem po3nr 4407

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 4406	. . 3 |- ( ( R Po A /\ ( B e. A /\ D e. A ) ) -> -. ( B R D /\ D R B ) )
2	1	3adantr2 1183	. 2 |- ( ( R Po A /\ ( B e. A /\ C e. A /\ D e. A ) ) -> -. ( B R D /\ D R B ) )
3		df-3an 1006	. . 3 |- ( ( B R C /\ C R D /\ D R B ) <-> ( ( B R C /\ C R D ) /\ D R B ) )
4		potr 4405	. . . 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 669	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 1004   e. wcel 2202   class class class wbr 4088   Po wpo 4391

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-po 4393

This theorem is referenced by:  so3nr  4419