Theorem po3nr 4038

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 RB))

Proof of Theorem po3nr

Step	Hyp	Ref	Expression
1		po2nr 4037	. . 3 |- (( R Po A /\ (B e. A /\ D e. A)) -> -. (B R D /\ D RB))
2	1	3adantr2 1063	. 2 |- (( R Po A /\ (B e. A /\ C e. A /\ D e. A)) -> -. (B R D /\ D RB))
3		df-3an 886	. . 3 |- ((B R C /\ C R D /\ D RB) <-> ((B R C /\ C R D) /\ D RB))
4		potr 4036	. . . 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 319	. . 3 |- (( R Po A /\ (B e. A /\ C e. A /\ D e. A)) -> (((B R C /\ C R D) /\ D RB) -> (B R D /\ D RB)))
6	3, 5	syl5bi 141	. 2 |- (( R Po A /\ (B e. A /\ C e. A /\ D e. A)) -> ((B R C /\ C R D /\ D RB) -> (B R D /\ D RB)))
7	2, 6	mtod 588	1 |- (( R Po A /\ (B e. A /\ C e. A /\ D e. A)) -> -. (B R C /\ C R D /\ D RB))

Syntax hints:  -. wn 3   -> wi 4   /\ wa 97   /\ w3a 884   e. wcel 1390   class class class wbr 3755   Po wpo 4022

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-po 4024

This theorem is referenced by:  so3nr  4050