Theorem po3nr 2839

Description: A partial order relation has no 3-cycle loops.

Assertion

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

Proof of Theorem po3nr

Step	Hyp	Ref	Expression
1		po2nr 2838	. . 3 |- ((R Po A /\ (B e. A /\ D e. A)) -> -. (BRD /\ DRB))
2	1	3adantr2 805	. 2 |- ((R Po A /\ (B e. A /\ C e. A /\ D e. A)) -> -. (BRD /\ DRB))
3		potr 2837	. . . 4 |- ((R Po A /\ (B e. A /\ C e. A /\ D e. A)) -> ((BRC /\ CRD) -> BRD))
4	3	anim1d 558	. . 3 |- ((R Po A /\ (B e. A /\ C e. A /\ D e. A)) -> (((BRC /\ CRD) /\ DRB) -> (BRD /\ DRB)))
5		df-3an 775	. . 3 |- ((BRC /\ CRD /\ DRB) <-> ((BRC /\ CRD) /\ DRB))
6	4, 5	syl5ib 206	. 2 |- ((R Po A /\ (B e. A /\ C e. A /\ D e. A)) -> ((BRC /\ CRD /\ DRB) -> (BRD /\ DRB)))
7	2, 6	mtod 108	1 |- ((R Po A /\ (B e. A /\ C e. A /\ D e. A)) -> -. (BRC /\ CRD /\ DRB))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   /\ w3a 773   e. wcel 955   class class class wbr 2609   Po wpo 2829

This theorem is referenced by:  so3nr 2850

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ral 1641  df-v 1803  df-un 2040  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-po 2831

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